I
I
wI
I
I
I
I
I
Ie
I
I
I
I
I
f
I
ASYMPXOTIC EQUIVALENCE OF REAL
PROBAB:r,I'l'Y DISTRIBUTIONS
by
Sadao Ikeda
University of North Carolina
Institute of Statistics Mirneo Series No. 522
January 1967
This research was supported by the
U. S. Arrrry Research Office-Durham
Grant No. DA-ARO-D-31-124-G814
This note is based upon a series of lectures
gi ven by the author during Spring Semester 1966;
it contains and refines the results of previous
works ["1,2,3,47, and includes some new results.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
I
I
i
CONTENTS
re
I
I
I
I
I
I
I.
1
INTRODUCTION
1
DEFINITIONS OF ASYMPTOTIC EQUIVALENCE OF PROBABILITY
DISTRIBUTIONS AND ASYMPTOTIC nIDEPENDENCE OF A SYSTEM
OF RANDOM VARIABLES
1.
Preliminaries.
2.
Definitions of stronger notions of asymptotic
3.
18
4. Notions of asymptotic independence of a syatem
of random variables.
II.
20
IMPLICATION RELATIONS OF NOTIONS OF ASYMPTOTIC
EQUIVALENCE
5.
Some properties of a sequence of random variables
in the case of equal basic spaces.
I
I
I
7.
I
III.
8.
23
Implication relations of certain types of
asymptotic equivalence in the general case.
26
Implication relations of certain types of
asymptotic equivalence in the case of equal
basic spaces.
33
Implication relations of notions of asymptotic
independence.
4h
TYPE (I,m) ASYMPTOTIC EQUIVALENCE AND THEm USE IN SOHE
PROBLEIvlS OF ASYMPTOTIC nIDEPENDENCE
e
v
9.
Some properties of type
(I,~)
asymptotic
equivalence and type (I,m) asymptotic independence. 47
I
I
13
Definitions of weaker notions of asymptotic
equivalence in the case of equal basic spaces.
6.
Ie
7
equivalence in the general case.
Ie
I
I
Page
SUMNARY
, 10.
Type (I,lB )asymptot ic equivalence cri teri a.
11.
Some applications of type (I,m) asymptotic
independence.
IV.
55
58
TYPE (I,8) ASYMPTOTIC EQUIVALENCE
12.
Some properties of type (I,8) asymptotic
equivalence.
60
ii
Page
13.
Properties of type (l,S) asymptotic independence.
14.
Measurable transformations preserving type (l,S)
66
asymptotic equivalence in the general case
15.
I
I
~
Measurable transformations preserving type (l,S)
asymptotic equivalence in the case of equal
I
basic spaces.
I
16. Type (l,S) asymptotic equivalence of marginal
random variables.
REFERENCES
83
97
I
I
I
I
_I
I
I
I
,
I
I
I
_I
I
I
I
P
I
I
I
I
I
I
Ie
I
I
I
I
Summary
Several notions of asymptotic equivalence together ,rith corresponding
notions of asymptotic independence for real probability distributions are
introduced and some of their properties are discussed.
These notions seem to
have 'ride applicability in the study of asymptotic approximation problems.
This work is a part of a construction
of a general theory of asymptotic
equivalence of probability distributions which the present author desires to
complete, and many unanswered questions are left open.
Introduction
Let us consider the following questions:
Suppose that a sequence of random variables, {X) (S=1,2, ..• ), converges
s
in law to some random variable Y as s tends to infinity. Let { c }
(i)
s
(s
= 1,2, ••• )
and put
N
X
s
(s
and {d )
s
= c s Xs
+ d
= 1,2, •.• )
N
s
and Y
s
be any given sequences of real numbers,
= c sY +
sometimes an asymptotic distribution of
N
d
S
for every
s.
Y has been called
s
N
X
Then, in what sense are these
s
two sequences of random variables approximate asymptotically as s tends to
infinity?
(ii)
Let {X.)
~
(i
= 1,2, ••• ,
N) be a system of elementary coverages obtained
I
by a random division of the interval [0,1) into N + 1 sub-intervals.
I
I
form of
Ie
N tends to infinity.
I
This is
not an jindependent system of random variables as is evident from the functi::mal
their joint probability density function.
when N increases.
Let us nOVl consider the case
It is easily noticed that any subsystem of fixed size of the
above system forms an independent system in some stronger sense asymptotically as
This suggests to us the following setting of a
question~
I
,
Let
n = n(N)
be a positi ve integer for each
choose any subsystem of size
n,
( X.)
(i
~
N such that
= 1,2, ••• ,
n->
n) say.
00
as N
-4
co,
and
Then, in order
that this subsystem forms an independent system in any given sense' asymptotically as
(iii)
N ->
00
,
how large could be the value of
n?
The following question concerns a central lirrd_t property of a dependent
system of random variables.
(X~)
(i = 1,2, ••• ,n),
s
~
tend to infinity with
Suppose we are given a system of random variables,
for each positive integer
s.
s, where
n
s
is assumed to
If this is an independent system for each fixed
then the system would have a central limit property:
s,
Under suitable conditions
s
there would exist sets of real numbers, { c.s } and {d.},
i=1,2, .•. , n , for
~
~
s
ns
each s such that
(c~l X~l + d~ ) converges in lavl to the standard
L:
l
i=l
We set forth a question in the
n~rmal distribution
N(O,l) as s -> 00
follovTing way:
eI
Is it still true that the system has the same central limit
independence" in a certain sense, and if so, what is the vleakest such sense
as~nptotic
(iv)
Let
independence?
{X~ns)
=
(X~""'X~s)}
(s = 1,2, .•• ) be a sequence of ns-dimensional
independent normal random variables with mean vector 0 and variance-covariance
ng
matrix
I
(unit matrix).
Then,
L:
ns
i=l
central chi-square distribution with n
n
s
y. 2
~
is distributed according to the
degrees of freedom. Suppose now that
s
tends to infinity with s, and that for any given positive integer n the
sequence of marginal random variables,
converges in law to
= (Yl ,.·.,
->
00
Yn )
as
(X~, ••• , X~)}
(s
~
sn; n Sn
s -> co. Is it possible in this
ns
2
ns
2
situation to say that two random variables
L: (X~)
and
L: Y., are
i=l
~
i=l l
distributed approsimately the same in some meaningful sense asymptotically as
s
Yen)
{X(n) =
I
I
I
I
I
t
proper"cy even if we replace the "independence" of the system by "asymptotic
of
ee
? n),
I
I
I
I
I
I
I
_I
?
2
I
I
I
,-I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(v)
Zem)) j
Let
(s
= 1,
2, ... ) be a sequence of
k+m
=n
dimensional random variables, Where k, m and n are assumed to be fixed
independently of
s.
Then for each
s
the marginal distribution of
has the cumulative distribution function
Hs (z (m)) •
Zem)
J
R(k) Fs (z (m) I X(k)) dGs (X(k) ),
where
G (x(k)) stands for the cumulative distribution function of X(k).
s
Let, further,
(s = 1,2, •.. ) be anotha:' sequence of random variables
which is "asymptotically equivalent" in some sense to the sequence
= 1,
(s
2, ••• ),
N
as
s
N
d GS(y(k))'
Gs(y(k))
->
•
Hs(Z(m))
= fR
Fs(Z(m) IY(k))
(k)
s
being the cumulative distribution fUnction of Y(k) ,
00
If we define
(X(k)}
then this gives a cululativ1distribution fUnction of a certain random variable,
NS
Z( m) say.
Possibly under some conditions
equivalent in some sense as
s ->
~
and
s
Z(m)
would be asymptotically
• Then, what types of asymptotic equivalence
s
s
could be expected between X(k) and Y(k)'
s
s
and Z(m) and Z(m) ?
To give answers for these questions it is necessary to introduce notions
of asynwtotic equivalence between two sequences of random variables, which
are defined independently of the existence of limiting distributions.
For most of the problems in which two sequences of random variables are
required to compare, we are given a sequence of events, {E } (s
s
= 1,
2, ... )
say, specified by subsets of euclidean spaces, together vdth two sequences of
random variables, (X ) (s
s
= 1,
are asked to examine whether
2, ••• )
Xs
P
(E )
s
to each other for large values of s.
and (Y ) (s = 1, 2, .•• ) say, and we
s
Y
and P s (E ) are sufficiently "close"
s
It should be noted that the dimensions
of underlying euclidean spaces are not alWays the same.
\'1e shall" call the above
type of problems the asymptotic approximation problem in general.
For a given
sequence
= 1,
2, ••• ), it is sometimes required to find another
sequence
= 1,
2, ••• ) which has the required closeness of P s(E ) to
s
Y
3
x
P s(E ) for given Ets or for any sequence of subsets, each of which belongs to
s
s
a certain class of subsets of the underlying space.
This problem will also be
included to the asymptotic assprximation problem.
As usual, two kinds of approximation error may be considered for the
asymptotic approximation problem stated above:
error",
y
p S(E )
s
One is the so called "absolute
Xs
and the other is "relative error", IP (E)-
I pXs (E s ) _pYs (E s ) I,
X
1/ p SeEs) ; the latter
s
will be useful for asymptotic evaluation of
X
IX)
an infinite series of probabilities,
E
a. P
s
(E .).
i=l
~
s~
These investigations naturally give us, for constructing a general theory
of asymptotic equivalence, concepts of basic spaces (underlying
~uclidean
spaces), basic classes (underlying sequence of classes of subsets of basic
spaces), and of two types of asymptotic equivalence, type I and II, corresponding
to absolute and relative error consideration.
The author developed in [1] a theory of type I and type II asymptotic
equivalence choosing
~-finite
measure apaces as basic classes.
These notions
of asymptotic equivalence appear too strong for some applications:
In fact,
the weaker, type I, turns out to be a notion of convergence in convergence
theory which is equivalent to that of "in the mean" convergence.
It is well
knovm that the notion of "in law" convergence plays an important role in
convergence theory, and some of the problems the author was required to answer
suggested necessity of introducing some weaker notions of asymptotic equivalence,
for example, one that is almost equivalent to an asymptotic equivalence theoretic
version of "in law" convergence.
In this note we shall confine ourselves to the case where the basic spaces
are euclidean, and the parameter involved is integral valued.
There is no.
special reason for this, except that it is more direct for application to
problems in usual statistics.
One could extend easily the idea developed
4
I
I
---I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
se
I
I
I
I
I
I
Ie
I
I;
I
I
I
I
I
Ie
I
here to the case of abstract basic spaces, as was done partly in [1].
In Chapter 1 we intend to give definitions of sereral types of asymptotic
equivalence and corresponding notions of asymptotic independence for real
probability distributions.
notions are exhibited
Some of the implication relati. ons between these
in Chapter 2.
Contents of Chapter 3 are essentially the
restatement of the results already given in [1] with some new results.
In Chapter
4 a special type of asymptotic equivalence which is important
for many applications is discussed. Main substances the author intends to
include in a general theory of asymptotiC equivalence are as follows:
(a)
Unconditional and conditional implication relations of different types
of asymptotic equivalence.
This problem is of importance not only for theoretical
completeness but also for practical applications of the theory.
(b)
Some suitable criteria for each type of asymptotic equivalence in
terms of characteristic functions, probability density functions, cumulative
distribution functions, or of any other easily confirmable quantity.
indispensable for applications of the theory.
This is
Unfortunately, for most of the
types of asymptotic equivalence, we have not sue ceded yet in finding such
criteria.
(c)
To determine a class of measurable transformations of random variables
which transfer any given type of asymptotic equivalence of the original sequences
to another given type of asymptotic equivalence of the reSUlting sequences.
This problem seems to have a wide applicability if any solution can be found.
In this note we give answers for this problem in a few cases.
There are some problems in the directions the author wants to extend the
present '\vork.
Throughout the present work, the dimensions of the basic euclidean
spaces are assumed to be constants.
However, for some cases of statistical
5
I
I
procedures it would be desirable to assume the dimensions
to be stochastically deterruned in some manner.
Any
of basic spaces
pre-investigation has
not been done yet in this problem, but the author hopes some fruitful results
could be obtained and the range of application of our notions of asymptotic
I
equivalence could be widened.
Presentation of a new concept of asymptotic independence of a system of
random variables, whose definition will be given in Chapter 1, seems to introduce
another aspect of problems different from the usual ones in the field of central
limit theorems, as was simply exemplified in the question (iii) stated in the
beginning of this section.
The author guesses it is an interesting work to
investigate these problems in connection with the usual results which have
already obtained in the indicated field.
The other problem the author intends to do is to find out a field of
applications in the theory of stochastic processes.
In this connection
Dr. R. M. Meyer has recently found that a notion of "mixing" for strictly
stationally stochastic processes is well expressible by our notion of
asymptotic independence of a suitable system of random variables, and this
m&~es
it easier to check the mixing condition in some practically common
situation.
To find further such applications is left to future investigations.
The author wished to express his thanks to all members of the Statistics
Lepartment for their help under Which the author has spent significant days
of the first year of his visiting.
The author's appreciations are specially
devoted to Professors N. L. Johnson, W. J. Hall, J. T. Runnenberg and to
Dr. R. M. Meyer for their helpful discussions and comments Which were given to
this work.
--I
Thanks are also devoted to Professor I. M. Chakravarti who has
given friendly encouragements to the author.
6
I
I
I
I
eI
,
,I
I
I
I
I
I
_I
I
I
I
--I
I
Definitions of Asymptotic Equivalence of Probability Distributions and
~totic
1.
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Independence of a System of Random Variables
Preliminaries
In this section we shall give some necessary notations and remarks about
a basic space, basic class and a family of probability distributions.
For any given positive integer n, let R(n) be the n-dimensional Euclidean
space, and
~(n)
the usual Borel field of subsets of R(n).
Let us denote
the family of all probability distributions defined over the measurable space
(R(n)' eBen»~ by 3(R(n)' eBen»~' the members of which will be designated by
random variables, X(n)' Y(n)' ••• ' say.
A probability
mea~ure
for example, X(n) is distributed will be designated by
that two members of the family 3 (R(n)'
Further, let
v(n) be any
~(n»
P (n)
according which,
It may happen
have the same probability measure.
a-finite measure defined over the measurable
space (R(n)' ~(n»' and denote the family of all probability distributions
which are absolutely continuous with respect to the basic measure
~(R(n)' eBen)'
V(n»·
v (n) by
It is well known that for any member of this family
there correspondS a generalized probability density function with respect to
v(n)- "gpdf(V(n»" for short---in the usual manner.
Let C(n) be any given (non-empty) subclass of eBen)' and let
Yen)
X(n)
and
be any two members of 3(R(n)' eBen»~. For these two random variables,
let us define the following two kinds of quantities
(1.1)
and
7
I
I
x
p (n)(E
(1.2)
y
(n)
p (n) (E
using the convention
ea
)
(n)
-1
)
I
0/0=1.
Note that the first quantity defines a distance over the family ~(R(n)'
eB(n))
if we identify those random variables which have the same probability
measure over the class
C(n)'
x(~)
and
Y(n)bei~
=p
bability measure over C(n) if p (n)(E(n))
of the class C(n).
It is also noted that if
called to have the same pro-
(n)(E(n))
for every member E(n)
X(n) and Yen) belong to the family
Q)(R(n)' men)' V(n)) for a certain v(n) and have gpdf (V(n))'s
f(n)(Z(n)) and
g(n)(Z(n)) respectively, then the equality
(1,3)
2 dr(X(n)' Y(n); (jJ(n))
=
JI
f (n) - g(n)
I
( Z (n)
)}
This is shown as follows:
and
B(n)
=
Putting A(n)
~
{z (n): f (n) ( Z (n) )
=
{Z(n): f(n)(Z(n))
g (n) ( Z (n))},
we have
I
I
I
I
_I
d v (n)
R(n)
holds.
I
> g(n)
I
I
I
f
J
~
because both
LHS,
On the other hand, since
A(n) and B(n) belong to eB(n).
x
is equal to
8
P (n) (E
Y
(n)
)_p (n) (E
(n)
)
I
I
_I
I
I
I
•I
I
I
x
or P (n)(E(n»
y
- p (n)(E(n»
It is also easily verified that
and
I
for any E(n) in ffi(n).
I
I
x
p (n)(E
(n)
y
P (n) (E
y
)_p (n)(E
(n)
X
(n)
)_p (n) (E
X
) < P (n)(A
=
(n)
=
Y
)_p (n)(A
(n)
X
(n)
)_p (n) (:8
(n)
)
)
Since then we can choose, for any given e > 0, a
member E(n) belonging <B(n) such that
+
we have the following inequalities
LHS
<
I
I
(n)
Y
) < P (n) (B
I
Ie
vdth its complementary
set E(n)' we can easily see that
I
I
I
I.
I
I
for any E(n) in ffi(n)
=
RHS+2t.
9
e ,
I
I
.-
Hence (1.3) holds true.
For the quantity defined by (1.2), one can show easily that
I
(1.4)
Clearly
the roles of X(n)
and Yen) are exchangeable in this inequality, and
therefore the vanishing of the quantity
drr(X(n)'Y(n); C(n»
implies that of
~I(Y(n)'X(n); C(n»' and vice versa.
The following inequality vall also be checked easily:
(1.5)
or, more precisely
I
I
I
J
I
e'l
I
Now in the next place we shall give some of the subclasses of the class
)1(n)
where
set)
n.
a
i
t
be the class of all subsets which are of the form
s are real numbers admitting the values
when a i
=- ~ ,
i = l, ••• ,n;
E(n)
= R(n)
~
00
,
when a i
and E(n) =¢ (empty
=+ ~
,
i
= 1, .••
Clearly, the intersection of any finite number of subsets belonging to
10
I
I
I
I
I
I
_I
I
I
I
--I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
H(n) is again in
The class
three classes,
ft(n)'
Sen) is defined to be the set-theoretical union of the
n(n)' _itcn)
and Sen)'
f{(n)
is the class consisting
of all corqplementary subsets of those which belong to}((n)
the class of all subsets of the
where
a.~ and b ~ ,
and S(n)
represents
fb rm
i=l, ••• ,n, are all real numbers such that
b. < a., i=l, .•• ,n.
~
=
~
It is evident then that
(1.6)
and
I1cn)
C
Q(n)
C CB (n)
•
It is also well known that (a) the class U(n) consists of all finite unions,
or equivalently,
of all finite disjoint unions of the members Sen)' i.e.,
U
-
N
L:
E
(n) - { i=l
(n)i
E(n)i
€
Sen)' E(n)iE(n)j:¢ (ifj);
i, j=l, ••
~,N
: N=1,2,...
}
and (b) for any member E(n) of Sen)' there can be found a positive integer
N and a set of members, F(n)l, ... ,F(n)N' of J{(n)'
the set E(n) and n, such that
N~ 2
n
Ie
I
where
11
and
both depending only on
I
I
ea
(1.8)
for any measure over(R(n)' ffi(n»' where ci's are constants which take
the
value J. or -1 depending only on E(n) and n.
The follovling statement is also wellknown:
V(n) over (R(n)' ffi(n»
For any O"-finite measure
and for any member E(n) of ffi(n)' the v(n)-measure
of E(n) is given by
co
V(n) (F (n»);
F(n)i
€
E(n)
C
i~l'
F(n)i,
G(n)' i=1,2, •••. }
for which we can take F(n)i's Imltually disjoint.
In the final place, a 'remark
~dll
be given on the notion of an indep-
endent system of random variables.
Let, as before, X(n)= (Xl'
x2,·.·Xn )
be a member of ~(R(n)' ffi(n»'
Corresponding to a decomposition of the space R(n) in the form
R(n)= R(n ) x R(~)X ••• X,R(nm)' n l +••• + nm = n,
l
we have a decompostion of the variable
X(n)= (X(n ), X(~)'.'.'X(nm»'
l
"There X(n.) belongs; to 3«R(n.)' ffi(n'j) for each i.
J.
random variables,
{
J.
Consider the set of marginal
J.
X(n )'" "X(n )
, of the above random variable. We
l
m
intend to call such a set of random variables a system of random variables
and m the size of the system.
{X(nl)""'X(~) }
According to the usual definition,
is said to be an independent system of random variables
if it holds that
pX(n)(E) = ~ pX(ni)(E
),
(n)
i=l
(ni )
for every product E(n ) x••• x E(n ) = E(n)' E(n.)
1
m
J.
(1.10)
12
€
8(n.)' i= 1,2, ••• ,m.
J.
I
I
I
I
I
I,
el:
I'
I
I
I
I
I
I
_I
I
I
I
,e
I
I
I
I
I
I
Any sUbsystem of an independent system of random variables is also
independent, but the inverse is not necessarily true.
2.
Definitions of stronger notions of asymptotic equivalence in the general
-case
In the first place we shall state the situation under which our notions
of asymptotic equivalence are defined.
(X~,
,x~
(y~,
,Y~
Let {x(n)=
...
)} (s=1,2, ••• ) and {Y(n )=
...
)}
s
s
s
s
(s = 1,2, ••• ), be two sequences of random variables, for "1hich X(n ) and
s
s
Y(n ) belong to a(R(n ),ffi(n » for each fixed value of the parameter s. The
se~ence
of Euclideansspace:, { R )]- Cs = 1,2, ••• ), will be called the
Cns
sequence of basic spaces. We shall call the case where the dimensions n
Ie
are identical to some positive integer n for all values of s the case of
I
I
I
the usual notions of convergence are defined in the former case.
t
I
I
I
equal basic spaces, and. the case of unequal basic spaces otherwise.
case of unequal basic spaces is such that n
Note that
An important
tends to infinity as s -+ 00.
Corresponding to the above sequence of random variables, let us consider
a sequence of classes of subsets of basic spaces, {C(n )} (s = 1,2, •.• ),
s
which will be called the sequence of basic classes, where for each s C (n )
s
is assumed to be a subclass of ffi (n ).
s
Under this situation, we shall give two notions of asymptotic equivalence
as follows:
DEFINITION 2.1.
Two sequences of random variables,
{X<ns )}
(s
= 1,2, •.. )
and { Y(n ) } (s = 1,2, ••• ), are said to be asymptotically equivalent in the
s
sense of type (I, C) _"ASEQ(I,C) II , for short _
as s -+ 00, and are denoted by
(I,C), (s -+00 ),
(2.1)
Ie
I
s
s
13
I
I
_I
if it holds that
(2,2)
Two sequences in the above definition are said to be
DEFilUTION 2.2.
asymptotically equivalent in the sense
for short _
(2.3 )
I
I
I
I
as s -+
s
X(n )
type (II, C ) -
af
"ASEQ(II,C)",
and are denoted by
00,
(II, C) , (s -+
to
),
s
if it holds that
(2.4)
d
In
II
(X
s
~
)
c
) -+ 0, (s -+ 00 ) .
(n )
s
[1] these types of asymptotic equivalence were
(n s ' (n );
s
the earlier work
I
defined by takingffi(n ) as C(n ) in the above definitions for probability
s
a-finite measures over abstract basic spaces, and were called type I and type
II
asym~ptotic
I
s
distributions which are absolutely continuous with respect to any given
equivalence respectively.
_I
It is noted that the second definition
2.2 is consistent by the inequality (1.4), though the left hand side of (2.4)
is not symmetric with respect to the two random variables involved.
The above definitions work, of course, in the case of equal basic spaces,
too.
In this case we can also consider the notions of convergence corresponding
to the two types of notions defined above, which will be stated in the
DEFINITION 2.3.
A sequence of random variables, {X(n)}
(s
follo~Qng.
= 1,2, ••• )
is s.aid to converge in the sense of type (I,C) to Y (n)' or converge (I,C)
to Y(n)'
in short, as s
(2.5)
X(n)
-4
00 ,
and is denoted by
-+ Y(n) (I,C), (s -+
00
),
if it holds that
(2.6)
dI(X(n)'Y(n);
DEFINITION 2.4.
C(n»-+
0, ( s -+
c:> ) .
The sequence in the shove definition is said to converge ...
14
I
I
I
I
I
I
I
II
I
I
I
,e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
in the sense of type (II,e), or, converge (II,e) to Yen) as s ~~, and is
denoted by
Yen) (II,C), (8
-+10 ),
if it holds that
Y
8
dII ( X(n)'
(n)'.C (n) ) -+ 0 , ( S -+ 00 ) •
(2.8)
Four definitions given above provide a large amount of notions
of asymptotic equivalence and of convergence, by specializing the sequence
of basic classes { C(n ) }, (s = 1,2, ••• ) or C(n) in the definitions to
s
any specific sequences of basic classes. Among those only several types of the
notions will be investigated in this note, without any special reason
except that they seem to be familiar to us, which will be given in the next
place.
We take especially the following five sequences of basic classes in
the case of unequal basic spaces %
{a (n )} (s=1,2, ••• ),
s
{
H(n)} (S=1,2, ••• ), {S(n )} (s=1,2, •.. ),
{q(n)} (8=1,2: ••• ), and {m(n )} (:=1,2, •.. )
s
s
whose definitions have been given in the preceding section.
In the case of
equal basic spaces, it is assumed that each of these turns out to be a single
class:
){(n)' Sen)' a(n)' q(n)' andm(n)'
By these specializations, we have
two kinds of stronger notions of asymptotic equivalence, each containing
five ,types of notions, and the corresponding notions of convergence in the case
of equal basic spaces:
type (I,lt), (I,S), (I,a), (I,q), and type (I,m);
type (II,~), (II,S), (II,a), ( II, ~), and type (II,m).
Mutual implication realtions of these notions will be investigated in
the later section.
In the final place of this section, we investigate some of
the equivalent conditions for type (I,C) and type (II,e) asymptotic equivalence,
which would be helpful
to the understanding of these notions.
15
I
I
The following lemma gives equivalent conditions to type (I,C) asymptotic
equivalence in the general case.
LEMMA 2.1.
Type (I,C) asymptotic equivalence of the sequences,
X(n )} (s=1,2, ••• ) and
{Y(n)} (s=1,2, •.. ), is equivalent to each
{
s
s
one of the following conditions.
(a)
For every sequence of subsets, { E(n ) }
(s=1,2, ••• ), such that
s
s
E(n ) belongs to C(n ) for each s, it holds that
s
~ns}
s
(2.9)
(b)
-+~).
I/fns} (E(n )} - P
(E(ni } -+ 0, (s
s
There exists a sequence of POSit1ve numbers,
qs } (s=1,2, ••• ),
~
depending only on the sequence of basic classes, { C(n
s
that q -+ 0 as s -+
(S=1,2, ••• ), such
and
to
S
)J
S
I
(2.10)
X
s
P (n s ) (E
(n )
~
)
s
(s=1,2, ••• ), with E(n )
s
for every sequence of subsets,
belonging to C(ns)
q, (s=1,2, ••• ),
s
for each s.
PRoOF: Evidently (a) is not stronger than (b).
It is also clear
that the condition (b) is necessary and sufficient for X ) N Yen)
en
(I,C), to prove the sufficiency of the condition (a), it s~ffices t3
merely point out that, if the sequences are not A8EQ(I,C), there exists
a subsequence [Sf} of [s} such that
s'
s'
X
Y
,
(n ')
s'
(n s ' )
P s
(E(n ')) - P
(E(ns,))1 -+ q, (8'-+")
s
for some q > 0, which contradicts (2.9). This proves the lemma.
In the case of equal basic spaces where n = n for all s, the
s
above two conditions reduce to the following:
(a)'
For every sequence of subsets, [E
belonging to the class C(n)' it holds that
en
)} (s = 1,2, ..• ),
-,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
16
I
I
I
Ie
(2.11)
(b) ,
I
I
I
I
I
I
It holds that
X
Ip
(2.12)
(n)(E(n»
-
P (n)(E(n»I
0, (s
~oo)
Parallel results to the above lemma. hold also for type (II, C)
asymptotic equivalence as will be stated in the following
( s
LEMMA 2.2. Type (II,C) asymptotic equivalence of tX(n ) J (s=1,2, .•. )
(
}
s
and tY(ns) (s=1,2, ••. ) is equivalent to each of the following conditions.
1
(a)
It holds that
1
pY(n )(E S )
s
(n.)
for every sequence of subsets,
~~(n
L
s
I
~ 0,
(s
~oo
),
)}S=1,2, ••• ), with E(n ) belonging
s
to C(n ) for each s.
(b)
There exists a sequence of positive numbers, {qs} (s=1,2, .•• ),
such that qs
~
S
Ip
(2.14 )
0
as s
~
X
)
(nS)(E s
(n )
s
10
and
~
_ p (nS)(E s
)1 ~ q .
s
(n )
s
for every sequence of sUbsets, {E(n )} (s=1,2, ••. ), with E(n ) belonging
s
s
to C(n ) for each S.
s
In'the case of equal basic spaces, these conditions turn out to the
following
(a) ,
0,
Ie
I
~
uniforIlll)x: for all E(n) belonging to C(n)"
Ie
I
I
I
I
I
I
I
~
S
17
(s
~
00
),
I
I
for every sequence of subsets, .{ E(n)
(b) I
}
(s = 1,2, .•• ), belonging to C(n).
It holds that
S
X
p (n)(E
(2.16)
(n)
)
....y£>
1
p (n) (E
(n)
)
I
~
0,
(s
~ co
)
uniformli" for all E(n) belonging to C(n)"
3.
Definitions of weaker
~otions
of asymptotic equivalence in the case of
equal basic spaces
As was seen in the last two lemmas of the preceding section, type (I,C)
and type (II,C)
asymptotic equivalence can be regarded as notions of asymptotic
equivalence based upon the uniform (over the basic classes) approximation
of two probability distributions to be compared.
In the present section we introduce some weaker types of asymptotic
equivalence in the case of equal basic spaces, basing upon the set-wise (over
the basic class) approximation.
Let, as before,
{X(n)} (s = 1,2, ••• ) and {Y(n)}
be tHO sequences of the members of
a
(s = 1,2, ••• )
(R(n)' m(n»' and let C(n) be any non-
empty subclass of ill (nr
First we shall give the following definitions.
DEFINITION 3.1.
{X(n)} (S=1,2, ••• ) and {Yen)} (s=1,2, ••• ) are said
to be asymptoticalli" equivalent in the sense of tyPe «I,C»- "ASEQ«IJC»~'
. in short - , as s
~
eo,
and are denoted by
s
X(n)
s
~ Y(n)
(
(r,c»,
(s
~co ),
_I
I
'I
I
I
J
I
_I
I
I
I
I
I
I
I
_I
18
I
I
I
I_
if i t holds that
....
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(S .... IO)
0,
for every subset E(n) belonging to C(n)"
DEFINITION 3.2. {X(n)} (s=1,2, .•• ) and { Y(n)
J (s=1,2, •.. ) are
said to be asymptotically equivalent in the sense of type «II,C»,"ASEg, « II, C) ) ~I in short -
(3 •3)
s
y (~)
S
X(n)
-
as s .... 10, and are denoted by
«II,C», (s .... 10),
if it holds that
S
X
(3.4)
p (n)(E(n»
_
1
I ..
0,
(s .... 00
)
yS
P (n)(E
(n)
)
for every E(n) belonging to C(n)' where we use the convention %
If we consider the case when Y(n)
f
S
= l.
are all identical to some fixed
distribution Yen) belonging to J(R(n),m(n»' the above definitions give the
corresponding notions of convergence.
DEFINITION 3.3.
{X(n)} (s=1,2, .•• ) is said to converge in the sense
of type «I,C», or "converge «I,C)g' in short, to Yen) as s .... 10
,
and is designated
by
s
X(n)
.... Yen)
«I,C», (s .... 10
),
if it holds that
(3.6)
X(n) DEFINITION 3.4.
Y~n)
(S .... IO).
«I,C»,
t X(n) }
(S=1,2, •.• ) is said to converge in the sense
of type «II,C», or "converge «II,C»~' in short, to Yen) as s .... 10
designated by
X )
Cn
.... Y(n)
19
«II,C», (s .... 10),
,
and is
I
I
_I
if it holds that
(3.8)
X(n) .... Yen)
«II,C», (s -+0:1 ).
By specializing the class C(n) to each of the classes
}1(n)' Sen)'
U(n)' q.(n) andCB(n)' we get five types of weaker notions of asymptotic equivalence
and the corresponding notions of convergence in the case of equal basic spaces.
Unfortunately, hmvever, the weakest type of convergence defined above, type
«I,X», is still stronger tInt that of in law convergence for some cases:
In fact, if we take, as the class C(n) in Definition
3.3 above, the class
i1(n)(Y(n»' formed by deleting from H(n)an those menbers ,·rhich' contain at
Yen)
least one discontinuity point of Yen)' i.e., a point whose P
-measure is
positive, then the corresponding convergence gives the same notion as in law
convergence.
type «I,M»
It is clear that if Yen) is of the continuous type, then our
convergence is exactly the same notion as in law convergence.
We shall give an improved result for this later.
4. Notions of aSYmPtotic independence of.a system of random variables
Notions of asymptotic equivalence given in the preceding two sections vnll
be used to define those of
asym~ptotic
independence of a system of random variables.
Let { X(n )} (s=l, 2, ••• ) be a sequence of random variables taken from
s
J(R(n ),CB(n }) for each s. Corresponding to the direct product expression of
s
s
each basic space in the form
R(ns)s= R(m~)X••• x R(~ )'
vre shall decompose
X(n) in the form
s
(4.1)
s
= ns,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
20
I
I
I
Ie
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
where
ks(~ 2)
{m~,. "'~}
and
may be dependent on s even in the case of
equal basic spaces, and x~(
s) belongs to ~(R(mS),ffi( ms» for each i.
J. m
i
i
i
Now we shall consider a system of random variables of size k formed
s
by all the marginals of (4.1), i.e.,
(4.2)
{
~( ~s)'. "'~s (~s
s )}
It is well known that for each s there exists a member of;; (R(n )' ffi(n »
s
s
(4.3 )
such that the system of marginal random variables
•
forms an independent system, and Y~C s) is identically distributed with X~C s)
J.~
J.~
for each i.
Let {C(n)} (s=1,2, ••• ) be any given sequence of basic classes such that
s
C(n ) is a subclass OfmCns)fOr each s.
s
Under this situation we give, firstly, stronger notions of asymptotic independence
in the general case.
DEFINITION 4.1.
A system of random variables (4.2) is said to be
asymptotically independent in the sense of type (I,C), or
as s
"ASIN(I,C)~1
in short,
if it holds that X(n)
Yen) (I,C) as s -41C1. Type (II,C)
s
s
asimptotic independence is defined in a qUite similar manner.
-41C1,
N
The following definition gives weaker types of asymptotic independence,
in the case of equal basic spaces.
DEFINITION 4.2.
A system of random variables (4.2) is said to be
asymptoticallY independent in the sense of type «I,e) ), or "ASIN( (I,e) );'
21
I
I
in short, as s -+ eo, i.f it holds that X(n)
«II,C»
asymptotic independence is also given in a similar manner.
_I
By these definitions we have various types of notions of asymptotic
independence.
I
I
I
I
Among those, we consider in the present note only several
types of notions, corresponding to the notions of asymptotic equivalence
considered in the preceding t"ro sections, Le., type (I,ft), (I,S), (I,G),
(I,Q), (I,m);
(II,~),
(II,S), (II,G), (II,q), (II,ili), in the general case,
and «I,,Al,», «I,S», «I,G», «I,q», «I,m»;
«II,ft», «II,S», «II,G»,
«II,G», «II,m», in the case of equla basic spaces.
It sometimes happens that the whOle system is not asymptotically independent,
but its sybsystems are asymptotically independent in a certain given sense.
The following definition will be useful for some of such cases.
Suppose that we are given a system of random variables of equal dimensions
(4.4)
4.,.
The system (4.4) is said to be asymptotically independent
(n ) in the sense of type (I,C),
s
or
I;'~)-ASIN(I,C)~I in
short, as s -+eo,
if every subsystem of size n s ' {X~l(m)' ••• ,X~ (m)}' is ASIN(I,C) as s -+CIO.
ns
Quite similarly, we can define (n s )-ASIN(II,C), in the general case,
and (ns )-ASIN«I,C»
or (ns )-ASIN«II,C»
_I
I
where m may be dependent on s, s=1,2, ••••
DEFINITION
I
I
in the case of equal basic spaces.
I
I
I
I
I
I
_I
22
I
I
I
I_
I
I
I
I
I
I
II
Implication Relations of Notions of Asymptotic Equivalence
5.
Some properties of a sequence of random variables in the case of equal
basic spaces.
It is iIlIpODtant to investigate the implication relations among the notions
of asymptotic, equivalence defined in the previous part and the eqqivalence
conditions, under which two different types of asymptotic equivalence are
mutually equivalent.
The purpose of the present section is to give two sorts of properties
for a sequence of random variables in the case of equal basic spaces, which
are useful to derive equivalence conditions for the notions of asymptotic
Ie
equiValence.
I
I
I
I
I
I
I
random variables.
The first one is a sort of stochastic boundedness of a sequence of
Let {X(n)} (s=1,2, ••• ) be a sequence of
variables.
For this sequence let us give the following
DEFINITION 5.1.
if for any given
€
{X(n)} (s=1,2, ••• ) is said to have the property B(S),
(>0) there exist a member of S(n)' B(n)say , whose closure
being compact, and a positive integer s
r,
p (n)(B
(5.1)
for all s
f::
s
(n)
) >
such that
o
€
o
In a similar manner as this definition, of course, we can give the
property B(e) for any subclass C(n) of <B(n)' which contains at least one subset
whose closure being compact.
Ie
I
~~dimensional random
23
We shall list some of the results on property B(S) in the following
LEl~
5.1.
(a)
Type «I,S»asymptotic equivalence brings over the
property B(S), i.e., if' XCn ).... ~n) «I,S» as s -+ 00 and { X(n) } (S=1,2, ... )
has the property B(S), then {Yen)! (s=1,2, ... ) has the same property.
(b)
Property B(S) is brought over by any type of asymptotic equivalence
which is stronger than or equivalent to type «I,S».
(c)
Let C(n) be any subclass of'm(n) which contain the class S(n).
Then, hlO properties, B(S) and B(e), are nru.tually equiValent.
(d)
If'the sequence {Xen)} (s=1,2, •.• ) converges. «I,S.»
to some
limiting distribution, then the sequence has the property B(S).
(e)
A sufficient condition for { Xen)} (s=1,2, •.• ) to have
the
property B(S) is that, for any given decomposition X(n)=(X~(~), ••• ~(~»,
k and {
~, ••• ~}
{X~(m.)}
J .
(f)
margina1s,
is
being independent of s, every sequence of margina1s,
(s=1,2, ••• ), has the property B(S), j=l, ••• ,k.
If {X en )} (s=1,2, ••• ) has the property B(S), then any sequence of
{X(m)=(X~l' ••• 'X~m)} (s=1,2, ••• ),
arbi~rarilY fixed
has the property B(S), where m
independently of s, while the choice of { i l ,· •• ,im}
out of {l, ••• ,n}may depend on s.
(g) If all the one-dimensional marginals of XCn) have finite ITeans and
variances uniformly for all s, then {X
en
)} (S=1,2, ••• ) has the property
B(S).
The proof of this lemma is straightforward from the definition and will
be omitted.
Another property is a sort of absolute continuity of a sequence of random
variables.
Let C(n)be any subclass offfi(n)' for which the following definition
I
I
_I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
is meaningful:
24
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(s
}
A sequence of random variables, tX(n)
(S=1,2, ••• )
DEFINITION 5.2.
is said to have the property C(C), if for any given
positive number 8
~
> 0, there exist a
and a positive constant s such that
o
for every E(n) in C(n) with
€ ,
for all s
€
s, where
o
~(n)
~(n)(E(n»< 8
designates the usual Lebesgue measure over
(R(n),m (n»·
Identifying the class C( ) in this definition vnth S( )' a
Q( )
n
n
(n)'
n
orffi(n)' we have the properties C(S), C(G), C(Q) or c(m), respectively. Note
that
CO() is meaningless, because
any member of the class ~n) has infinite
Lebesgue measure except for the null set.
We shall list below SOIre of the natures
of these properties, which are direct consequences of the definition.
LEr~ 5.2.
(a) If
~(n)}
(s=1,2, .•• ) has the property C(C), and
C(n) ~ crn)' then the seClUence has the property C(C*).
(b)
Property C(C) is brOUght over by type «I,C»
asymptotic equivalence,
and also by any type of asymptotic equivalence which is stronger than
OT
equivalent to type «I,C».
(c) If {XCn )} (S=1,2, ••• ) has the property C(C) and C(n) is anyone
of the four classes mentioned above, then the sequence of any marginals,
{ XS(m)=(r1. , .•. ,X~l. )} (s=l, 2, ••• ), has
1
independently of
s~ while
~he
same property, vThere m is fixed
the choice of {i l , •.. , i m} out Of{ l, ••• ,n} may depend
on s.
(d)
condition:
Property C(C) of { X(n)}
For any given
a positive number
8
€
(s=1,2, ••• ) is equivalent to the following
> 0 and any given positive integer N, there exist
and a positive integer s , both depending on
o
N, and such that the conditions
N
~(n)( i~l E(n)i) and E(n)i
€
25
C(n), i=l, ••• ,N
€
and
I
I
_I
imply jointly that
S
X
N
p (n) (U
i=l
for all s
~
E
<
.)
(n)~
€
so' where the sets { E(n)i }, 1=1, .•• ,N, mayor may not be mutually
disjoint.
(e)
Suppose that {X(n)} (s=1,2, ••• ) converges «I,C»
to Yen)' an
n-dimensional random variable which is of the continuous type, i.e., which is
absolutely continuous with respect to the Lebesgue
measure~
Then, the sequence
has the property C(C).
6.
Implication relations of certain types of asymptotic equivalence in the
general case
In this section we shall consider the implication relations of stronger
notions of asymptotic equivalence defined in Section 2, where we have defined
two kinds of stronger notions, type (I,C) and type (II,C), assuming that
{ C(n ) } (s=1,2, ••• ) is any given sequence of basic classes such that
s
) for each s.
C(n ) S CB
ens
s
The followeing lemma gives some of the general implication relations of
these notions of asymptotic equivalence.
LEMMA 6.1.
(a)
(I,C) is a weaker notion than (II,C).
(b)
Let {C(n ) } (s=1,2, ••• ) and {C(n )} (s=1,2, ••• ) be two sequences
s
s
of basic spaces such that C I (n )S C(n ) for each s. Then, (I,C I), or (II,C'),
s
s
is weaker than (I,C), or (II,C), respectively.
Hereafter, we shall use the symbol
-+
to denote an implication relation.
For example, the assertion(a) of the above lemma will be designated as (II,C)
-+
(I,C), or equivalently as (I,C) +- (II,C).
26
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
By the lemma stated above, then, we have the follovling diagram of implication
relations among those types of notions of asymptotic equivalence which have
I
I
I
I
I
I
been given in Section 2:
t
(II,,Ai)
Of course, these implication relations are valid for any pair of sequences of
random variables without any additional condition.
Now, we shall show the following theorem.
THEOREM 6.1.
(I,~)
:; (I,G), i.e., these types of notions are mutually
equivalent.
Ie
I
I
I
I
I
I
I
(1,11.)
(6.1)
PROOF From (6.1), it suffices to show that
(6.2)
. (I,~)
For any given
~
€
(I,a).
>
S
0 and each s, there exists a member of~(n )' B(n )
s
and there can be found two covering of B ),H. s= fH(n )i} (i=1,2, ••. ) and
Cn s
L s
s
} (j=1,2, ••• ), consisting of members of the class G(n )' such that
)(, s
= lK(ns)j
s
r
-
(
Ie
I
s
say, such that
27
I
I
_I
where it can be assumed without any loss of generality that each of the
both coverings consists of mutually disjoint members of U(n)O
s, £,s
= { L(n )k} (k=1,2, ••• ) be the product covering of
~he
Let, for each
above two
s
coverings, which is obtained by rearranging all subsets of the form
H(ns)i n K(ns)j' i,j,.=1,2, ••• , in any suitable manner.
Note that L{ns)k
belongs to the class G(n ) for each k, and Len )k n L(ns)k' = ¢ (k4k').
s
Then,
s
putting L(n ) = f L(n )k' we have
s
s k=l
(6.4)
~It
s
is noted that L(n ) does not necessarily belong to U(n )"
s
s
Since, for each s, there exists a positive integer N=N(€, s) such that
N
s
s
where 'ole have put LN(n ) = l: L( )k' it follows from
k=l
ns
s
(6.4) that
+
from "'lhich we obtain
4€,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
28
I
I
I
I_
s
because LN(n ) belongs to
s
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
Since
E
a.(n
)for each s.
s
is arbitrary, (6.5) implies that
for every s, from which the theorem follows.
Type (I,S) asymptotic equivalence appears to be weaker than that of type
(I,G ), or equivalently of type (I,m), in an important case:
In fact, the in
la"[ convergence of the standardized binomial variable vri th parameters n ani
P to the standard normal distribution as n -+ 10 is a convergence of type (I, S),
as 'rill be seen later, but it is not of; type (I,G).
Now, in the next place, we shall consider the implication relation between
(I,S) and (I,I() in the general case.
The following example shows that these two types of asymptotic equivalence
are not necessarily mutually equivalent.
EXAMPLE 6.1.
Let Xl"'" Xs be a random sample cf size s drawn from the
uniform distribution on the interval [0,1), While Y~, ••• ,Y: be that from a
distribution with p.d.f. given by
p
' lis, if -l{z<O,
(z): l-l/S, if O~z<l,
{ 0, otherwise.
s
Then, for a set E(s) in S(s) defined by
e(n)=[O,l) x••• x[O,l) , (8 times),
one can observe, putting X(s)=(xl,···,Xs ) and Yes) =(Y~, ••• ~), that
X
p (s)(E(s»
yS
- P (8)(E(s»
I = 1-(1-1/8)S
29
-+
l-l/e, (S
-+00 ),
I
I
which shows that X(s)
-
Y(s) (I,~~, (s -+-), does not hold.
On the other hand, it is easy to see that
which implies that these two sequences are
ASEQ(I,~
as s
-+-.
A sufficient condition, under which (I,S) and (I,JL) are mutually
equivalent, is given by the following
~ K, s = 1,2, ••• , for some constant K
s
independent of s, then it holds that
THEOREM 6.2.
(6.7)
If n
_I
I
I
I
I
I
I
el
(I,S ) ~ (I,~
with X(n )' Y(n ) belonging to ~(R(n_)'~(n )) for each s
s
s
~
s
In particular,
the above equivalence is true in the case of equal basic spaces.
The proof of this theorem follows from (6.2), the assumption of
30
I
I
I
I
I
I
I
-I
I
I
I
I_
I
I
I
I
I
I
for every s.
As for the implication relation between (I,q) and (I,8): we have the
following result, whose proof is easy and will be omitted.
THEOREM
If the sequences{X(n )
... ) and {ytng )1(S=1,2) ... ))
s
s
are such that, for each S,X(n ) and ytn ) are of the continuous type, i.e.,
s
s
are absolutely continuous uith respect to the Lebesgue measure over (R(n ),
s
ffi(n )), then it holds for these sequences that
s
(6.9)
(I,Q) ~ (I,S).
Now, we proceed to the implication relations of tYl)e IT
The following theorem is due to
THEOREM
6.4
D~.
R. M. Meyer.
For any pair of sequences of random variables it holds
that
(6.10)
PROOF.
~
(II,m)
To show
(II, it)
~
(II, S) •
(6.10) it suffices to prove that
(6.11)
Since any member of
~(n
) is expressed as a union of a finite number of
s
the members of S(n ) for each s, it is easy to see that, for any given
member B(n ) of I(:s) and any given e > 0, there exists a covering of B(n )'
s
s
{ E~ns)r (i = 1,2, ••. ) say, consisting of mutually disjoint members of
~(n )' such that
s
Ie
I
series of
notions of asymptotic equivalence.
Ie
I
I
I
I
I
I
I
~S=1,2
6.3
31
I
I
_I
for each s.
Since the asymptotic equivalence (11,0) of both sequences of random
variables considered above implies the existence of a sequence of positive
numbers
-~
~
q} (s = 1,2, ..• )
SUC11
S
that q
s
.-+0 as s
~to,
and
I pX~nS)(E~n )i) - pY(ns)(E(n )i) I:;: qs pY(ns)(Etns)i),
s
s
for every i and s, ..le have
Hence we get
Xs
yS
P (n s ) (B(n » - p (ns)(B(n »
I
s
for each s.
s
e
Since
inde3)endent of
e ,
I < qs PyS(ns)(B(ns »
s
)e ,
is arbitrarily small, and B(n ) is also arbitrary and
s
the above inequa l i ty implies that
s
I :;: qs pY(n)s (Bns
s
s
X( ) s
Y()
p ns (B(n » - p ns (B(n»
(6.12)
=q
+ (2
s
s
s
for each s, which shows that the sequences are ASEQ(II,ffi).
) ,
This proves
(6.11) and hence the theorem.
It is quite easy to see the follO'.dng
s
COROLLARY 6.1.
s
If both X(n ) and Y(n ) are of the continuous type,
s
s
then it holds that
(6.13 )
(11,115)
~ (II,ll)
~ (II,i..»
~ (II,y).
Type (II,yJ is a weal(er notion than type (II,i..», as l'lill be seen in the
foHol-ling
EXM·:!PLE 6.2.
is given by
z
Ps ()
Let X be one-dimensional random variable whose p.d f
s
=J S /(S-1),
La,
if a ~ z < 1-1/s ,
other'iTise,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
Ie
I
I
I
I
I
I
for s = 2,3, •.• , while
X converges to
s
(II,S) as s
I
I
I
I
Ie
I
Y in the sense of type
(II~~),
Then clearly
but not in the sense of type
because it ahmys holds that dn(Xs"Y;S(l))= 1, s ... 1,2, ...
Summarizing the results thus obtained we have the f:::lH:::lving diagram of
implication relations
bet~~en
stronger types of noti:::lns :::If asymptotic
equivalence in the general case, where the symbol
I
--.
unconditional implication relation, while the symbol
I
designates an
'---~
I
does a con-
diti:::lnal, for which a sufficient condition has been given in this section.
(6.14 )
Ie
I
I
I
"-10,
Y be the uniform distributi:m on (0,1).
The following questions are interesting and left open.
(i)
To find out useful equivalence conditions under which (I,S) and (I,m)
are mutually equivalent.
(ii)
To find out equivalence dontitions for (II,u) and
(iii)
(II,~).
To find out equivalence dontitions for (I,ci) and (II,S), for (I,~),
and (II,p,U, and for (I,m) and (II,ffi).
7.
Implication relations of certain types of asymptotic equivalence in the
case of equal basic spaces
In the case of equal basic spaces, we have defined, in addition to
stronger types of notions, some weaker types of notions of asymptotic
equivalence.
The results obtained in the preceding secti:m can equally be
applied in the present case, that is, the implication relati:ms shown by
(6.14) hold in the case of equal basic spaces too. Furthermore the mutual
equivalence of (I,;;;) and
(I)~
is unconditionaL
Equivalence conditi::ms,
hovrevcr, for (II,S) and (II,J-(,) to be mutually equivalent are stHl.
unkn01-lIl
even in the case of equal basic spaces, as is seen from Example 6.2.
It is evident from definitions that
((I,C»
for any C(n)
(I,C) is a
included by eB(n)
str~nger
notion than
,and the same is true for (II,C)
and ((II,C».
Now vre shall consider the impl.ication relations of weaker types of
noti~ns.
It is clear that the following diagram of implication relations
is obtained:
.
~
((I,Q»
1
~
((I,CB» ---1 ((I,U» ~ ((I,S»
(7.1)
r
((II,m»
-4-
-4
((I ~»
iii
T
((II,~»
((11,0» ~ ((II,~)
~((II,q»~
It is shown easily that
It always holds that
THEOREM 7- L
(7.2)
((1,0,,»
i=?
((I,S»;:::: ((I,,v)
I
I
_I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
«1,4».
Ie
I
I
I
I
I
I
Since any given member of
a (n)
is expressed as the union of a finite
number of mutually disjoint members of Sen) , and any given member of ti(n)
is expressed in a form by a finite number of members of
, we have a
N
K
= ~
~
Cij E(n)ij ,
i=l
j=l
for any member A(n) of £ten) , where c ij (=+l),N and K are constants
A(n)
depending on A(n) , and E(n)ij are the members of f\(n).
Xfn)
P
YS(n)
(A( »-P
(A(
n
n
Hence
N K
XS(n)
YS(n)
Ip
(E( ) .. )-p
(E( )'j)
= i=l j=l
n ~J
n ~
» 1< ~ ~
I
for each s, from which (7.3) follows.
«I,e»
is stronger than «I,~» as will be seen in the following
EXAMPLE 7.1
Let Xs and Ys be one-dimensional random variables
Whose p.d.f. IS are given respectively by
Ps (z)
__ {2/S' if i~<i+l/2, i=O, 1, ••• s-l
o , othervrise,
and
qs(z) =
for s=1,2,...
2/s, if
i+l/2~<i+l, i=O,l, ••• ,s-l
{ 0, otherwise,
Then, it is clear that {Xs ) (s=1,2, ••• ) and{Ys) (s=1,2, •.• )
are asymptotically equivalent in the sense of type
sense of type
«1,18»,
as S,
00+10.
Ie
I
~(n)
formal expression such as
Ie
I
I
I
I
I
I
I
To prove (7.2) it is sufficient to show that
PROOF.
35
«I,lA. »,
but not in the
I
I
In the next place, we shall show the following
THEOREM 7.2
f1
( (II,£it) )
EBOQF.
_I
It always holds that
«II,S».
For any given A(n) of a(n)' there exist a positive integer
Nand N members of S(n)' E(n)l, ••• ,E(n)N say, such that
N
A()
n
U E
i=l (n)i,
=
from vThich it follows inunediately that
- 11 ~ ~ I pX(n) (E (n)i)
13
-i=l
-
11 ,
p (n)(E(n)i)
for each s.
This implies that «II,S» -.
follows from
(7.1).
«II,CL», and the theorem
The following two example s shows in turn that « II, S» and « II, to\, »,
and «II,CB» and «II/l»
EXAMPLE 7.2
Let Xs
are not necessarily mutually equivalent.
and Ys
be one-dimensional random variables whose
p.d.f. being given respectively by
l-l/s, if -l~z<o,
l/s, if O~<l,
0, otherwise,
ps(z)
={
~(z)
= { 1/(23\
if.O~~,
o , o{nerm.se,
and
for s
= 1,2,...
l-l/s
if-l~<O,
Then, it is checked easily that these two sequences are
ASEQ( (II, f(), but not ASEQ( (II, S», as
s -,10.
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
36
I
I
I
Ie
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Let
EXAIID?LE '7. 3 '
wh~se
X
s
and Y
be one-dimensional random variables
s
y.d.f. being given respectively by
~'(2S)
{
P (z)
s
=
if 0 ;?
Jtherl'lise,
(l
q (z)
s
=
u
-:
~...-s
)/(2s), if 21
~_
(1 +..2:-)/(2S), if 2i
~+s
+ ',_;j '~
= 0, 1 , .
< z <. 2i; i
= 1,2,
z <
1
2~...
.,
s- 1 )
,. ,s ,
o , ot11er'vise,
for s
= 2,3, • ..
Then, it is evident that these two sequences are
.
ASEQ«II,S)), or equivalently ASEQ«II,£»), as
however, the set
S-+IO.
If ',Ie take,
E defined by
co
E =,E L~i,2i
].=0
+ 1),
vlhich is a member of UJ(l) , then it holds that
X
p SeE)
for each s.
= 1/2
Y
s-l 1
i
P seE) =,E:2S (1--r+s) ,
and
~=o
Hence
s-l...L
'+
~ s
=i=o
E
-+
log2, (s -+ 10
).
which shows that the above sequences are not ASEQ«II,8)).
Now, we return to the implication relation of «I,a)) and «I,ili)),
and give an equivalence condition under which they are mutually equivalent.
THEOREM 7.3.
{ X(n)}s
= 1,2, ••• )
If both of the sequences of random variables,
and{ Y(n) j<s= 1,2, ... ) , have the property C (m), and
at least one of them has the property
«I ,ill ))
72
«I,(l.)) •
Ie
I
z < 2s,
37
B(S), then it holds that
I
I
As we stated in Section 5, it is seen from the assumption
PROOF.
that both of the sequences have the property B(~).
_I
Hence, in order to
prove the theorem, it suffices to show that the asymptotic equivalence
«I,~))
of the sequences in the sense of type
implies that
... 0, (s
"'-0 ),
for any member E(n) of (\)(n) whose closure is compact.
Let ~(n) be the usual Lebesgue measure over (R(n),m(n))'
any given 0 > 0, there exists a covering of E(n)'{ A(n)ij<i
Then for
= 1,2., ••• )
consisting of mutually disjoint members of a(n)' such that
For any given
e > 0,
we can chose, then, by the property C(CB) of both
sequences, such a value of 6 in
S
(7.6)
pX (n) (E(n)) - pX (n) i~l A(n)i)
I pY()n (E( n ))
for all s
~ s~
S
10
_ Py (n)p::
A( ),)
l=l
n l
I< e
,
I >e
,
for some positive integer
For the same values of
f~
_I
that
S
s~
•
and 6 as above, there exists a positive
integer N depending on 6 such that
eo
,
and hence
(7.8)
,
and
for all s > sit for some
-
0
Sll
0
•
say,
I
I
I
I
I
I
I
I
I
I
I
I
I
_I
38
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
From (7.7) and (7.8) it now follows that
and
I PY(n)(E (n) )
for all s ~ So' where So
~
- pY(n)( ... A
i~l
= max(s~,s;),
I
<2 ~ ,
from which
(7.5)
easily follows,
and the theorem.
From (7.1), Theorem 7.1 and Theorem 7.3 it immediately follows that
COROLLARY 7.1.
Under the same conditions as in Theorem 7.3, it holds
that
The following theorem gives a sufficient condition for (( I,q)) and
( (I, S)) to be mutually equi valent.
THEOREM 7.4.
{ Y(n)J (s
= 1,2, .•. )
If one of the sequences Of{
= 1,2, .•• )
and
((I,S)) .
(7.10 )
PROOF.
xfn )} (s
has the properties C(S) and B(S), then it holds that
From Theorem 7.1 and the diagram (7.1) , it follows
innnediat.ely that
((I,G))
~
((I,S)),
without any additional condition.
Hence, in order to sh::>,;,! (7.10), it
suffices to prove
((I,S))
~
((I,Q)).
Ie
I
.)
(n)l
39
Suppose now that the sequences are ASEQ«I,S».
Then, the properties
I
I
_I
C(s) and B(S) of one of the sequences are both brought over to the other
sequence, and therefore it necessarily follows, under the condition of the
theorem, that both of the sequences have the properties C(S) and B(S).
Let G(n) be any given member of Ci (n) , whose closure being compact
Then, for any given 6 > 0, there exist positive integers N andN' (N
~
N'),
and a set of mutually disjoint members of S(n)'{ E(n)ij(i = 1, •.. ,N, .. ,jl') say,
such that
N
Z
N'
N'
i=l
E().C G( )
n
~
-
Z
C
n - i=l
E().
n
1
.
and 1.1 (n) ( Z E E( ). )<0
i=N"+l n 1
where l.1(n) is the Lebesgue measure over (R(n), CB(n»'
For any given
e>
s
X
p(n)(G(»
n
0, then, we can choose 6 above so small that
_p
I
xtn) N
(~E(
),) <e
1=1 n ~
,
_I
,
I
I
I
I
I
I
I
and
for all s > s
-
0
xf )
Ip
n (
for some So depending only on
)
yS
G(n) -P
(n)(G
(n)
)
N
X
1<i=l
Z Ip
S
e,
N
from which it follows that
~
N
(n)p: E
) pJ.~n)(Z E
)
i=l (n)i i=l (n)i
I + 2e.
for .all s > s •
=
0
(7.11) now follows from (7.12), and the proof of the theorem is
completed.
In the final place, vTe shall prove the following theorem, which is
useful for many cases of application.
I
I
I
I
I
I
,
_I
40
I
I
I
I_
s
(Y(n)}(s
= 1,2 .•. )
( 7. 13 )
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
If one of the sequence~ (Xfn)}(s
THEOREM 7. 5•
has the properties c(~ ) and B(S), then it holds that
( I, S )
«I, S ) ) •
~
To prove the theorem it suffices to show that
PROOF.
Suppose the sequences are ASEQ«I,S)).
have the properties C(S) and B(S).
sequences are not ASEQ(I,S).
Then, both of the sequences
Suppose, on the contrary to (7.14), the
Then we can assume, without any harm in the
proof beloH, that there exisc a positive constant p and a sequence of
members of S(n),{E(n)} (s
= 1,2,~~.)
say, such that
for all s.
By the property B(S) of both sequences, there exist a positive
integer So and a member B(n) of Sen)' whose closure being compact, such
that
s
X(n)
P
(B(n))f l-P/2
for all s> s o •
=
for all s > s.
Y~n)
and P
Thus, putting
(B(n))~ l-p/2
s
A(n)
=
s
E(n)nB(n) for each s, we have
s
Here, it is noted that E(n) 's are members of S(n)'
0
NovT, since the closure of A(n) is bounded uniformly for all s, there
exist a subsequence{ s'}(s'-4~ ) Of~S} and a sUbse: A(n) of R(n) such that
(7.17)
lfm A(n) (=
s -4~
l~m.inf A~n)
=
S -4~
Ie
I
= 1,2, ... ) and
ljl
l~m.sup Afn)
S -4~
) = A(n) .
For this set A(n)' there can be found a member E(n) of Sen) such that the
~
symmetric difference E(n)
A(n) is contained in the boundary set of E(n)'
I
I
_I
and hence the closure of E(n) and that of A(n) are identical, and, of
course, they are compact.
By the property C(~) of both sequences, then, for any given
e > 0,
n
there exist a set of members of S(n),(F(n)i}(i : 1, .. N), N = 2 , and a
,
positive integer So such that
N
E(n)~ A(n) 'E i~lF (n)i
and
for all
5
>
Therefore, it holds that
= 5'o'
xf~) (E
p
(n)
) _
pxf~) (A
(n)
)
,
(7.18)
5'
5'
pY(n)(E
for all s' ~
- 5'a
(n)
) - pY(n)(A
5"
o
=
S"
a
I< e
(7.17) it follows that there exists a
such that
A(n)
for all r >
)
•
On the other hand, from
positive integer
(n)
~
,
N
Afn) <i.U
F( n ).~ ,
~=l
Hence, if s' >
max (s',
5 "), it holds that
=
0
a
s'
,
s
X(n)
X ),
p
(A(n» - p (n (A~n» <e
I
s'
I pY(n)(A(n»
(7.18) and (7.19) imply that
,
-
pY(n)(A(~»
I< e
I
I
I
I
I
I
el
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(7.20 )
for all s' > max (s' s ")
=
0'
0
which contradict (7.16), because the first member
'
of the right-hand side of (7.20) tends to zero as
s~~.
This proves the theorem.
\'fuence we immediately have the following
COROLLARY 7.2.
Under the same candition as in the above theorem
it holds that
(I, S ) ~
(7.21)
(I, ~
~
« I, CL) )
~ « I, 0> ) )
~
« I, ~) )
~ « I,
(n.
From Theorem 7.5 it is straightforward that
COROLLARY 7.3.
If a sequence of random variables, [X(n)}(s
= 1,2, ... )
converges in law to a certain distribution, Y(n)' which is of the continuous
type, then
this convergence is of type (l,8)
He summarize the results thus obtained t·o get the folloHing diagram
of
i:~licatiJn
relations of ':Teater notions of asymptotic equivalence in the
case of equal basic spaces.
~
«l,(J»
___
~--~
~
«l,a»;:;:!.' «I,S » ~ «I,~»
----
«I,c.-»
(7.22)
t
«11,(1»
=:
---t
If'
t
«Il,(A,»
~
\
t;=
«II,S»
«IJ,q»
~ «II,~))
~
Some questions are left open for completing the above diagram of
implication relatians, among which it is interesting to find out sufficient
conditions (as weak as possible) for «I,S»
«II,M,», for «Il,a»
mutually equivalent.
and «II,S», for
and «Il,GJ», or for «II,S»
«I,~»
and
and «ll,M,), to be
,
8
I
I
Implication relation of nJtions of asymptotic independence.
For the notions of asymptotic independence defined in Section
_I
4, the
second sequence (Y(n )Hs = 1,2, .. ) is determined completely from t1'2 first
s
sequence (X(n )1(s = 1,2, .), whose marginals are defined to be asymps
totically independent, in a manner described in that section.
Hence, among the conditional implication relations of notions of
asymptotic equivalence obtained in the preceding two sections, those results
which have conditions imposed on the dimensions of basic spaces and/or on
either one of the sequences to be compared are transferred to the same implication
relations of notions of
~symptotic
independence with slight modification on
the statement of the conditions.
It is clear that all results on the absolute implicatiJu relations
obtained befqre are transferred to the
correspo~ding r~sults
'on those of.
.1
el
notions.of asyntptotic independence.
In the first place, we shall prepare the following lemmas:
s
S
s
( X ( s)
Let, as before, [X(n)
)>1 (s = 1,2,. ) be a sequence
1 ~ '''''Xk ( IS
s~
of random variables, whose marginals
s
=
(8.1)
for s
= 1,2,
s
(s
, are under investigation, while let Y(n)=
Yl(~),
yS
J.
s
ks(~ )
(s = 1,2, . . ) be a sequence of random variables such that the i-th margina1 S
is identically distributed with that of X(n) for each sand i; i = 1,.
and the set of margina1s of Y(n) corresponding to
,k
s
(8.1) is an independent
system for each s.
Under this situation, it is clear that
s
If (X(n)}(S
I
I
I
I
I
I
= 1,2,
4'4
) has property
B(S), then (Y(n)f
'
)
I
I
I
I
I
I
I
_I
I
I
I
(s
= 1,2,.,.) does the same propert,y.
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
From Lemma 5 2 (b), it foHOl-FS easily that
LEW-1A 8 2,
Suppose that the system of random variables (8 1) is
AfIN«I,C)) as s~~ , and the sequence (X(n)}(s
) has property
C(C), where C(n) is any subclass of Cli(n) for which the stated property is
well defined.
In
Then, the sequence [Y(n)l(s
= 1,2,
) has the same property C(C)
the general case, the fC)l1owing le,mma is easily verified:
LE~~ 8.3
If, for each s, X(n ) in (4 1) is of the continuous
s
s
type, the corresponding variable Y(n ) in (4 3) is also of the same type
s
NOvT, Theorems 6 1,6 2 and 6.4 hold for notions of asymptotic independence without any change of the statenent
Theorem 6 3 can be restated,
by Lemma 8.3, as follows:
COROLLARY 8
If X(n ) is of the ccntinuous type for each
1
5,
then,
s
for two notions of asymptotic independence, (I,o.) and (I,S), it holds that
(8 2)
(I,G)
~
(I,S)
It is also easy to see, corresponding to Corollary 6 1, that
COROLLARY 8 2
Under the same condition of Corollary 8 1, it
holds that
(8 3)
(II,G)
«II,s).
Thus we have the same diagram' as that of (6 14) on the implication
relations of notions of asymptotic independence in the general case:
~ (I,q) ...
~
(I,lEi ~ (I,ll)
(8.4 )
(II,m)
==
T
t·~
(I,S) <--~ (\~
(II,tL) _
(II,S) - 7 (II,~)
~(III,Q.(~
Ie
I
= 1,2,
45
In the case of equal basic spaces, we have the same diagram as above
except for one relation:
In this case, (1,5) and (I,t() are mutually equi·-
valent without any additional condition
As for the implication relations of weaker types of notions of
asymptotic independence in the case of equal basic spaces,
~ffi
have the
follovnng results by using Lemmas 8.1 and 8 2
First, corresponding to Theorem 7 4, we get
COROLLARY 8 3.
If the sequence [X(n)}(s = 1,2, . ) has both of the
properties C(S) and B(S), then it holds that
(8 5)
«I,Q))
~
«1,5))
From Theorem 7.5 we have
COROLLARY 8 4
If [X(n)}(s
= 1,2,
) has the properties C(S) and
B(S), then it holds that
(8 6)
(I,·S) ~
«I,S)).
Hence, we obtain the following diagram of implication relations of
weaker types of notions of asymptotic independence in the case of equal basic
spaces.
(8,7)
In concluding this section, it is noted that we have not any result
on equivalence conditions, under which type «I,ili.)) and type
asymptotic independence are mutually equivalent.
«I,~))
If we can prove the
result of Lemma 82, without the first condition that (8.1) is ASIN«I,C)),
we can obtain a corresponding result to Theorem 7 3
46
I
I
_I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
ill
Type (I,G})
9·
Asymptotic Equivalence and some of their Applications
Some properties of type
(I,~)
asyrrwtotic equivalence and type (I,ili)
asymptotic independence
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Notion of type (1,13) asymptotic equivalence defined in Section 2 is
3), to be equivalent to that of type I defined in the earlier
seen, from (1
paper
L-1J,
if we confine ourselves to Euclidean basic spaces and to the
family of all probability distributions which are absolutely continuous
with respect to any given a-finite measure over each basic space
In this section we shall sketch the fundamental properties of type
(I,CB) asymptotic equivalence, most of 'thich have been presented in
:"-1_'
) be
Let, as before, tX(n )}(s = 1,2,
) and (Y(n )1(s = 1,2
s
s
t"10 sequences of random variables, where X(n ) and Yen ) belong;, (R(n )'
s
s
s
Type (I,~) asymptotic equivalence of these two
ill (n » for each s.
s
sequences was defined by the condition
S
- py (ns)(E
(n s )
)
I
-. 0 , (s-. eo)
In the case of equal basic spaces where n
notion of type
(I,m)
s
= n for all s, this gives a
convergence, if we take a certain fixed distribution
s f
Y(n) instead of Y(n)s •
The following lemma gives equivalent conditions to type (I,ili)
asymptotic equivalence in the general case, which is a restatement of
Lemma 2.1.
LEMMA. 9.1.
The condition
(9.1)
for type (I,m) asymptotic equivalence
is equivalent to each one of the following conditions.
47
(a) For every sequence of sUbsets,[E(n )} (s
s
for each s, it holds that
(b)
=
_I
There exists a sequence of positive numbers, {qs }(s
= 1,2, ••• ),
depending only on fCB(n~ \ (s
(9.3 )
I
I
xtn)
Ip
y
= 1,2, .. ),
such that qs-+w (s -+w) and
S
s (Efn s » - p (ns ) (Etn »
s
I ~ qs
' (s
= 1,2, ..• ).
= 1,2, .•• ) with E(n )
€ ffi(n ) for each s.
s
s
Now, let (R(ms),CB(~Ii) (s = 1,2, ... ) be another sequence of
for any sequence (E(ns)}(S
Euclidean spaces with Borel fields, where we assume n
s
> m for each s,
=
s
Let further f(n ,m )(Z(n » be a measurable transformation with defining
s
s
S
space R(n ) and range space R(m ) for each s.
s
This transformation gives
s
us sequences of transformed variables,
s
= 1,2, ••• ,
belong to ;J(R(m ),Cl\m » for
s
s
where X(m )and y8(m )
s
s
each s.
Then it is immediate to see the following
THEOREM 2·1.
x(n)"" Y(n ) (I,m), (S-+IO ), implies that
s
S(m ) .... Y(m )(I,ili), (S-+IO).
s
s
The coverse is true if the transformation
s
fS( n ,m ) is non-singular for each s.
s s
The following is immediate from this theorem or the definition of
type (I,m) asymptotic equivalence itself.
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
48
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
COROLLARY 9. 1.
marginals of X(n ) and Y(n ) respectively for each s, where k s and the choice of
s
s
til, ••• ,ik} out of (l, •• :,n} may depend on s. Then, X(ns)N Y(n ) (I,Q),
s
s
s
(s-+to), implies that X(k )N Y(k )(I,ffi), (s -+to).
s
S
Next, we shall proceed to the asymptotic equality of moments of
asymptotically equivalent (1,03) sequences of random variables" LX(n )3 (s
.. , .
s
= 1,2, .•• )
and {,Y(n )J (s = 1,2, ... ) in the general case.
Let f(n ,1) (Z(n » be a
s
s
s
measurable transformation from R(n )
Put
s to R(l) for each s.
for each s.
Then, Theorem 9.1 assures us that if the original sequences are
Ai;)EQ(I,CD) as s-'7l>", then
LXs J(s = 1,2, ... ) and{Y s J(s
= 1,2, ... ) are also ASEQ (f,ili)
Now let us ask what conditions guarantee the symptotic equality of
expected values of these two sequences of induced random variables.
This problem will be turn out to the following:
Let
= 1,2,
[X }(s
s
,)
and (Ys } (s = 1,2, ..• ) be sequences of one-dimensional random variables, which
are ASEQ (I,(£,) as s-+ to , and let us denote the expected values of X and Y
s
by M(Xs ) and M(Ys )' respectively, for each s.
out the conditions under which it holds that
I M(Xs ) - M(Ys ) I -+
0,
s
Then the problem is to find
(s -+to).
This question is not a trivial one:
For example, the existence
of the expected values M(Xs ) . and M(Ys ) is not necessarily sufficient
(9.5), as is seen in the follo~dng example.
for
EXANPLE 9.1.
-
Suppose Xs and Ys be distributed according to the
normal distributions N(O,os2) and N(l,os2), respectively, for each s, where
os
~ eo as s ~ eo.
Then, by using Theorem 1. 4.2 of
is an extension of the relation
IEMr.fA
9.2
as
°
s ~eo.
(1.3).
Let X(n) and Y(n) be two ~mbers of 3«R(n),ffi(n))' and
let Pen) and Q(n) be the corresponding probability measures.
Then, it
holds that
I
= JRld(P - Q)I,
2 dI(X,Y;CB)
where we have dropped the suffices for dimension, and the right hand member
is given by
= lim
8
E€ r (R,m)
5
ro(R,ffi) being a partition of R consisting of (den~rable number of) members
of eB(n) whose radii are all less than 6.
For any given 6 and a partition
PROOF.
be subclasses of
P(E)-Q(E) >
L:
r (R,ffi), let
6
f
'lE-
5
and f
**
5
I
respectively
P(E)-Q(E) 1=
E€r5 (R,CL)
=
Then, we have
Ip(E)-Q(E) I + L: Ip(E)-Q(E)
EEf**
o
6
(P(F*) - Q(F*)) + (Q(F**) - P(F**)) ,
I:
E€[*
where F* and F** denote the unions of all members of
rt and [r respectively.
Since F* and F** belong to (lI(n]' it follows from the above equality that
(9.8)
6
r 6 (R,w) consisting of all such subsets E in r (R,m) that
° and < 0,
2 dI(X, Y;ffi) ~ R
I
d(P-Q)
50
I
I
I
I
I
I
I
_I
I P(E) -Q( ~) I
L:
~o
_I
["'l'J or Theorem 11 2 in
However, M(Xs ) =
and M(Y s ) = 1 for all s, which shows that (9.5) does not hold.
To anSvTer the above question, we prepare the following lemma which
later section, it is seen that Xs -Ys (I,ffi)
I
I
I
I
I
I
I
I
I
_I
I
I
I
I_
For any given
such that
2 dI(X,y~) < 2 (P(B)-Q(B»
(9.9)
I
I
I
I
I
I
I
I
I
I
I
I
Ie
I
+
e .
The first member of the right-hand side becomes
2 (P(B)-Q(B»
= (P(B)-Q(B»
+
(Q(B)-P(B»
, where B denotes the
complimentary set of B.
For any given partition r&(R,ffi), let ft(R,ffi) be the product
partition of r (R,ill) and tB,B}. Then, rt(R,m) can be divided into two
6
subclasses, V (B,ffi) and Vo(B,tE), which are partitions of B and of B,
6
respecti vely.
We then have
P(B)-Q(B) =
E
E£V (B,ffi)
(P(E)~(E»
E
and Q(B)-p(B)=
(Q(E)-P(E».
E£V (B,CB)
6
o
I
Ie
e > 0, there exists a member of ili(n)' B say,
Hence, from (9.9) and (9.10) it follows that
2 ~(X,Y;ili) <
(9.11)
t
Eeft(R,m)
I P(E)-Q(E)I
+
e ,
which yields
2 dI(X, Y;CB)
~ j R Id(P-Q) I
+
e.
(9.6) now follows from (9.8) and (9.12), which proves the lemma
It is evident that, when p(n) and Q(n) are dominated by some
finite measure v(n) over (R(n),m(n»'
(J'-
(9.6) becomes (1.3).
Now it is easy to show that
THEOREM 9.2.
Let [X }(s = 1,2, •.. ) and [Y }(s = 1,2, ••• ) be two
s
s
sequences of one-dimensional real random variables, which are ASEQ(I,ffi) as
Suppose that the second order moments of X and of Yare finite
s
s
uniformly for all s. Then, the expected values of Xs and Y are asymps
totically equal in the sense of (9.5).
S
~IO
•
PROOF.
This follows from the inequalities
I M(Xs )-M(Y
) I 2 =5
s
-
x2 ld(P -Q ) I
(1)
51 s s
JR
JR(1) Id(Ps-Q,s) I
2
< 2 (H(X S )
+ M(Ys2».dI(XS'YS;ll.) (1»
I
I
,
where M(X 2) and M(Y 2) stand for the second order moments of X and Y
s
s
s
s
resnectively,
It is interesting to ask a similar ouestion as above for the
asymntotic equality of
e~ected
values in the sense that
M(X )VM(Y) 000+ 0, (sooo+eo), which is left o-,en,
s
s
In the last half of this section, \<7e give some "'1ronerties :If
tyne (I,J_j) asymptotic independence.
Let (X(n )
s
= (X~(
S)""'X: ( s»} (s
~
s ~
s
= 1,2, ... )
be a sequence
I
I
I
I
I
of random variables, whose marginals
(9.14)
I
are subjected for investigating the asymptotic independence (I,m).
The
corresponding sequence (Y(n ) }(s = 1,2, •.• ) is given by the same as in
the beginning of Section 4.
s
Suppose now that there is a sequence of random variables,
(Z(n )
s
-,
= (Z~(~ ), ... ,Z~s(~
»}
(s
= 1,2, ... ), whose marginals
s
(Z~(~), ..• ,Z~s(~ )}
(9.15)
s
form an independent system of random variables for each s .
Under this situation, I'le can show the following
THEOREM 9.3.
(a) If the sequences (X(n )} (s
= 1,2, .•• )
s
= 1,2,
{Z(n )} (s
s
.•. ) are ASEQ(II,CE) as S-+IO and the condition
k
(1 + ~I(X(n)' Zen ); CB(n»
s
s
s
s-+ 1,(s -+10), is satisfied"
52
and
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
then the system of random variables
in particular, n
for
s
are finite uniformly for all s, then it is sufficient
(9. 14) to be ASIN( l,CE) as s
PROOF...
-+ -, that
,
(a) Since (II,C~) is not weaker than (I,ll), it follows
from Lemma 2.1 (a) that
s
X
p (n ) (E s
I
(9.14) is ASIN(I,ili) as -+-, (b) If,
s
(n)
s
)
for every sequence{ E(n )} (s
s
= 1,2, ••• ),
each of Which being of the form
The assumption of the theorem implies that
k s t! s
II p i(mi)(E~ s» 1-+ 0, (s -+10) , o'r equivalently,
i=l
~(mi
I
Ie
I
I
I
I
I
t
I
Ie
I
Hence, it follows from
(9.18)
I
(9.16) that
XS
yP,
s ) - P (n s ) (s
P (n)
s (
E(n)
E(n) )
far any sequence
S
I
-+0,
( )
s -+10 ,
S
{E(n )} (s = 1,2, ••• ), each of Which belonging to the
s
direct product, ffi(n ) =ffi(~)X ••• ~(~)
s
s
**
Let, for each s, CB (n ) be the class of all finite disjoint unions
s
of subsets belonging to the class mtn )
Then it is clear that the class
s
** ).
. over the field eB(n
s
ili(n ) is the smallest cr-field
s
Furthermore, it is observed that
(9.18) holds true for every sequence
{E(n )} (s = 1,2, ..• ) with E(n ) belonging to
s
s
53
ili(~).
I
Hence the result (a) is sho..m by quite a similar proof to that of
The orem 6. 1.
(b)
If n
s
is finite uniformly for all s, it is easy to see that
the asymptotic equivalence (I,ffi) of { X(n )} (s
s
(s
= 1,2, ••• )
= 1 , 2 , ... )
l
"s ) }
an d J U(n
s
guarantees (9.17).
I
-,
I
I
Thus the theorem is proved.
The result (b) in the theorem can immediately be applied to get
the follovring
COROLLARY 9.2.
'converges (I,CE) to Y(n)
random variables,{
Suppose that { X(n)
= (Yl' .•• ,Yn )
X~, •.• ,X~}
= (X~, ... ,X~)}
as S-+«I.
(s
= 1,2,- .. ),
Then, the system of marginal
, is ASIN(I,ffi) as s-+ «I , if and only if
{Yl ,···, Yn} is an independent system.
Now', as before, we shall consider a measurable transformtion
-==s
-s
Let X(m ) and Y(m )
s
s
be induced random variables defined by
(9.4).
Then, the following theorem is an immediate consequence of
Theorem 9.1, which proof is omitted.
THEOREM
9.4.
If, for every s, the transformation is such that
a set of marginals ofytms ),{
~(;t~)'''''Y~s(t~)}, say,
forms an
s
independent system, then the corresponding set of marginals of X(m ),
s
"'~ (tS
{~(.tS)"
1
sh
) } is ASIN(I,CB)
as s
-+«1, under the condition that
s
the marginals of original random variable X(n )'
s
{X~(
s), ••• ,~ ( s )},
~
s ~
s
I
I
I
I
_I
I
I
I
I
I
,
I
_I
I
I
,_
I
I
I
I
I
I
I
Ie
is ASIN(I,Cl3) as s -+ 10
•
If, in particular, we take a projection as the transfornation,
the independence property of the system {
~(~)"'" ~s ({
)} is preserved,
s
which gives us the following
COROLLARY
9.3.
Any sUbsystem of asymptotically independent (I,m)
system of random variables is asymptotically independent (I,C\)
under the
same limiting process.
10.
Some criteria for type (I,m) asymptotic equivalence.
In this section, we shall give two kinds of criteria for type (I,m)
asymptotic equivalence, which have been presented in ~1-7.
Let { X(n )} (8 = 1,2, ••• )_and {Y(n )} (8 =
s
s
sequences of random variables with x{n ) and Yen) in
s
s
r
1,2,.~.)
be two
»
(R(n ), ffi(n ), v(n
8
8
S
I
I
I
Let us introduce the following quantities:
(10.1)
I
(10.2)
I
for s
I
I
Ie
I
d
V(n )
s
,
d v (n ) ,
s
= 1,2, ••• ,
'Which are called ttie "affinity" and the "Kullback-Leibler
mean information" respectively.
Note that the latter is a directed quantity,
and of c:>urse one can define an analogous quantity by interchanginging the roles
of X(n ) and Y(n ). In both of these definitions, it whould be understood
s
8
that the difference of carriers of gpdf' s are taken into account:
55
I
For example, in the definition (10.2), if v(n )(D(f(n »-D(g(n ») > 0,
s
s
s
then the quantity is defined to be infinity, while if v(n )(D(g(n »s
s
» and
D(f(n ») > 0, this contributes nothing to the quantity where D(f(n
s
s
D(gs(n s » designate the carriers of fS( n ) and gS()
n s respec t·1ve 1y.
s
Now, for these quantities the following is easy to verify (L-lJ).
LElv1ro1A 10. 1.
It holds always that
for each s.
From this lemma we immediately have the following two criteria
for type (I,m) asymptotic equivalence.
In order that{ X(n )} (5
s
(s
= 1,2, ••• ) are ASEQ(I,CB) as
(10.4)
p(X(n )'Y(n»
s
...
s~eo,
1,
s
= 1,2, ... ) and{ Y(n )}
s
it is necessary and sufficient that
(s ~..,),
and an error estimation in this case is given by
,..---~----
THEOREM 10.2.
In order that t'K) sequences in the above thec:>rem
are ASEQ(I,<B) as s ~"', it is sufficient that
(10.6)
0,
(s~ eo),
or
I(Y(n ):X(n»
... 0, (s ... eo),
S
s
where an error estimation is gi ven by
(10.7)
.~
-,
I
I
I
I
I
I
The proof of this lemma is omitted.
THEOREM 10.1.
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
(10.8)
In the earlier paper
I
I
I
I
I
I
THEOREM 10.3.
are ASEQ(I,~) as
(10.9)
(10.10)
L-IJ we
have also given the following criterion:
In order that{,X(ns)jks
S"'IO ,
=1,2,
) and{ Y(ns)jks
it is sufficient 'Mhat
pX(nS)(D(f(n »-D(gCn ») ... o,pY(nS)(D(gCn ».D(f(n »)-+ 0,
s s
H
s
s
ess.sup
f(ns/g(n ) - 1
-+0,
s
I
I
as s -+19 , where'ess.sup' is taken over D(f(n »nD(g(n » with respect to
s
s
\I
(n ) •
s
Any other criteria in terms, for example, of characteristic
functions, moment-generating functions, or of distribution functions
Ie
have not been available yet, and these are left open.
I
I
I
criteria.
Now, in the final place, we give a remark for type (I,ll) convergence
Let us consider the sequence{ X(n)
j<s • 1,2, ••• ) and a random
variable Y(n) in the case of equal basic spaces, where all of X(n) 's and
I
I
I
I
Ie
I
= 1,2, •.• )
PY(n)(E (n) ) , ( s....) ,
uniformly for aJ.l E(n) in m(n).
For this convergence, we have had a
familiar criterion
(10.12)
57
as s
I
I
~ 10.
It has been shown that our criterion (10.6), or (10.7) given in
Theorem 10.2 above is incomparable with the above criterion (10.12), Le.,
one cannot say that (10.12) is not weaker than, or not stronger than (10.6),
or than (1O • 7) •
In this section we shall show some applications of notion of
asymptotic independence (I,ili) , the last two of which have been seen in
We omit here the detailed calculations or proofs.
(a)
Suppose, first, that we are concerned with the Liapounoff
central limit theorem, Which states that if a system of one-dimensional
random variables
(l1.l)
{
X~, ••• , X~s}
, (with
is an independent system for each
l
(11.2)
with
of
n
/
s>S'3J
1
3
L ~.v
M{X?)
J.
x~-m~ = {~~)3
,
21/
l
i-l\'
= m~J.
ns~1O as s ~IO),
s, s = 1,2, ••• , and
n
s
Ls~cr.)
i=
the Liapounoff c.ondition
~ 0,
(s~ 10)
Var{X~)2
= {cr~)2
and the 3-rd order absolute moment
J.
J.
(provided that these are all exist), is satisfied, then
the standardized sum
(11.3)
:.
i=l
(X~ - m?J /
J.
I
I
I
I
_I
2
J.
n
I
I
n. Some applications of type (I,ill) asymptotic independence.
~1-7.
-,
I
I
I
I
s
Z
i-l
converges in law to the standard normal distribution N{O,l) as
s~ 10,
and
hence in the sense of type (I,S) by Corollary 7.3.
It is easily seen from Theorem 9.1 that this theorem still holds
true, even if the independence property of the system (ll.l) is replaced
by type (I,m) asymptotic independence property, because the condition (11.2)
58
I
I
I
_I
I
I
I
Ie
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
depends on (separated) properties of marginals, not on the inner structure
of the Whole system (11.1).
It is interesting to ask the following questions: (1
0
)
What is the
weakest notion of asymptotic independence of (11.1) which guarantees the
Liapounoff central limit theorem?
(20 )
How small could be the values of
1s for Which (1 s ) - asymptotic independence (I,m) of (11.1) assures us the theorem?
These questions are left open.
(b)
Suppose that we are given a system of elementary coverages,
{C } (i = 1,2, ••• ,N), Which is obtained by random division of the interval
i
[5,1) into N+l subintervals.
{C
il
Any sybsystem.of size n
0:
.ne whole system,
, ... C }, has the pdf. given by
in
(11.4)
f
(n)
(z
(n)
)
n
(l-E
r(N~n+l) i=l
=
r(N+l)
which is independent of the choices of (il, ••• ,i ).
n
From (11.4) it follows, putting n = 1, that each marginal has pdf. given by
(11.5)
f(z)
=N (l_z)N-l
, (0 ~ z < 1).
Then, Theorem 10.2 can be applied to get the following result:
The system (Cil (i
= 1, ••• ,N)
is (n)-ASIN(I,m) as N -+10, provided
that n = o(!ii).
(c)
Let Xi ~ ••• ~ ~ be an order statistics drawn from a one-
dimensional probability distribution of the continuous type, and let
!(n)
= (xl,···,Xn )
and X(m)
= (XN-m+l' ••• 'XN)
be joint distributions of n
lower extremes and that of m upper extremes, respectively.
Then by making use of Theorems 9.1 and 10.2 we can show that a system
of random variable, (!(n)'X(m)} , is ASIN(I,CB) as N -+ 10 ,provided that
n + m = 0(;; ); we shall say this as
I
totically independent (I,m) as N -+ 10. I
59
!(n) and X(m) are nmtually asymp-
I
I
1V
type (l,S) Asymptotic Equivalence
12.
Some properties of type (l,S)
as~totic
equivalence.
In this section we shall show some properties of type (I,S) asymptotic
equivalence specially in relation to the usual in probability convergence.
Firstly, it is clear from the definitions of in probability convergence and type (I,S) asymptotic equivalence that
LEr~~ 12.1.
ASEQ(I,S) as
S~IO
= 1,2, ••• )
If [X(n)} (s
,
and [Y(n)J (s
= l,a, ••• )
are
and if one of these sequences converges in probability
to a point c(n) = (cl' ••• 'c n ) as S~IO
probability to the same point as s
,
then the other converges in
~ 10 •
Note that this lemma still holds even if we replace 'ASEQ(I,S), in
the lemma by 'ASEQ((I,S»'.
Sometimes we meet with the following question: For a given sequence
f s
of random variables, tX(n)
'::S
(
)
12.1
..II.(n)
s ••• ,Xs) } ( s = 1,2, .•• ) , let us put
= (xl,
n
= (.clXl ,···, cnXns) '
S S
S
s
s
where c.~ are constants such that c.~
situation, it is asked that tx
are ASEQ,(I, S) as s ~ 10
Cn
~
)} (s
s
= 1,2 ••• ,
1 as s
~
10
= 1,2, ... )
,
for each i.
and lX{n)J (s
Under this
= 1,2, ••• )
•
An answer for this question is given by the following
THEOREM 12.1 If [X(n)}(s
B(S), then [X(n)}(S
PROOF.
= 1,2, ••• )
= 1,2, ... )
and tX{n)}(s
has both of properties C(S) and
= l,a, ••• ) are ASEQ(I,S) as S~IO.
First, it is noted that there is no harm in assuming that
s
c i > 0 for all i and s.
-,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
Ie
I
I
I
I
I
From property B(S) of lx(n»)(S
given
Ie
I
e ,
for all s > s •
=
0
Now, put
E(n)(al,···,an ) = [zen) = (zl,···,zn)1 zi > ai' i = l, ••• ,n) ,
for any real numbers a i ' i = l, ••• ,n • Then, clearly
s
I Pxtn) (E(n)(al,···,an»-Pxtn) (E(n)(al,···,an » I ~ PX(n) (A(n»'
(12.3)
for all s, where A(n) is the symmetric difference defined by
A(n)
= E(n)(a l , ... ,an )¢. E(n)(~/c~,
... ,an/c:).
Put, for each s,
Ie
I
I
I
there exist, for any
e > 0, a positive integer So and a member M(n) of Sen)' whose closure
being compact, such that .
s
X( )
(12.2)
p n (M(n» > 1 -
I
I
I
I
I
= 1,2, ••• ),
F
S
-
s
M
(n) - (n) nA(n)
Then, this is a disjoint union of at most n members of Sen)' whose closures
are all compact.
Furthermore, since M(n) in (J2 .2) is independent of s,
we can assume that there exist a positive number K, and a positive integer
s~
such
that if I a i
I > K for
some i, then the set Fen) is empty for all
s ~ s~ •
Hence, by property C(S) of (X(n»)(S
= 1,2, ••• ),
positive integer S"
such that
.0
X
(12.4)
S
p(n)(F s(n)
»e ,
uniformly for all (al ,. • .,an ) and for all 8 ~ 8~ •
From (12.2), (12.3) and (12.4), it follows that
dI(X(n)'X(n) ;~n» > 2
e ,
there exists a
I
for all s >
max(s 0 ,S',S"),
which proves the theorem, because type (I,u..)
=
0
0
,.....
and type (I,S) asymptotic equivalence are mutually equivalent in the case
of equal basic spaces.
The above result could be extended to the case of unequal basic
I
-,
I
spaces, though it is still an open question.
The following examples show that the conditions of the theorem are
not necessarily removable.
EXAMPLE 12. 1.
variance 1, for s
Let Xs be a normal random variable with mean ms and
= 1,2, ••• ,
X = c X , for each s.
s s
s
(s
with m -.. as s -...
Then c
s
s
-.1 as s-. 10
,
Put c
s
= 1-1/ms
, and
and the sequence Lx }
s
= 1,2, ••• ) has property C(S), but not B(S). It is easily seen that
p(X s
= lis)
=
Let Xs be a discrete random variable such that
and c s -. 1 as s-.-.
Xs = c sXs
, where c s > 1 for all s,
Then, lxs }(s = 1,2, ••• ) has property B(Li), but not C(S).
1, s = 1,2, .•• , and let
Again ,,,e can see that they are not ASEQ(I,S) as s -. to.
Hereafter in this section 1-re shall restrict ourselves to the case n = 2.
Firstly, we show the fol1ovring
THEOREM 12.2. Let [(X ,Y )l(s = 1,2, •.• ) be a sequence of twos s
dimensional random variables satisfying the following conditions:
(i) The first marginals, [Xs)(s
= 1,2, ••. ),
(ii) The second marginals, [Y }(s
s
to a certain constant c.
Then it holds that
I
I
both sequences are not ASEQ(I,S) as s -,00.
EXAl,1PLE 12. 2.
I
I
I
has property C(S).
=1,2, ... ),
converges in probability
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Since, for any given & > 0 and any given real number a,
PROOF.'
= p(x8
p(x +Y < a)
S8
< a-Y ,
S
I YS -c 1 > &)
+ p(x < a-Y ,
S
S
I YS -c I <= 0)
and
p(x < a-c-6) < p(X < a-Y , I Y -c I < 0) < p(X < a-c.O)
s
= s
8
8
= = S
p(X < a-Y , Y -<I I >6) < p( I Y -c I >6),
s
s
s
-'
S
I
we have
I p(Xx +Ys
(12.6)
< a)-P(X +c < a)
s
I $_ P(a-c-6
< X < a-c+o )+p(
- s
Iys -c I> {),
for each s.
By
the condition (i) of the theorem, there exist, for any given
e > 0, a value of 6,
(12.7)
&0
say, and a positive integer So such that
sup P(a-c-~ ~ Xs < a-c+Oo) <
- <
a
e
<-
for all s > s •
=
0
On the other hand, by the condition (ii) of the theorem, there
exists a positive integer
s~
such that
(12.8)
for all s > s' •
=
0
It follows from
(12.6), (12.7) and (12.8) that
sup I p(X +Y < a)-P(X +c < a)1 < 2
-<a<s S
s
e
for all s > max(s ,8'), which shows that
=
0 0 ·
Xs + Ys
XS + c (I,J'.), (s ....),
or equivalently (12.5}.
N
This completes the proof of the theorem.
The following corollary is an immediate consequence of this theorem.
COROLLARY l,2. 1.
(a)
Suppose that the two conditions, (i) and (ii),
of the above theorem are satisfied, and [X )(s
s
another sequence [Scs ) (s = 1,2, ••• ) as s ... -.
= 1,2, ... )
is ASEQ(I,S) to
I
I
Then. it holds that
(I,S) , (s.....).
(b)
(i')
In the above theorem, if the condition (i) is replaced by
[Xs )(s
= 1,2, •.. )
converges in law to a certain distribution, Z,
of the continuous type as s ... eel,
then it holds that
(12.10)
PROOF.
(b) This follows from the above theorem and Corollary 7.3.
Note that the classical convergence theorem due to Crarrer states
that if Xs converges in law to Z (not necessarily of the continuous type)
and Y converges in probability to a constant c as s... " , then X + Y
s
s
converges in law to Z + c as s......
s
If Z is of the continuous type, then
this reduces to (b) of the above corollary.
Now, we shall prove the follO'ving
Let [(X ,Y )}(s = 1,2, ••• ) be a sequence of twos s
dimensional real random variables satisfying the following t"lO conditions:
THEOREM 12.3.
(1) [Xs j (s = 1,2, ••• ) haS both of properties C(S) and B(S).
(ii) tys }(s = 1,2, ••• ) converges in probability to a certain non-zero
constant c.
Then, it holds that
X / y ,., X / c (1,8), (s.....).
s
s
s
(12.11)
and
y
(12.12)
PROOF'.
s
,., cX
s
(1,8), (s.....).
It is evident that y
(in prob.), and vice versa.
s
... c (in prob.) implies that l/Y ... c-1
s
Hence (12.12) follows from (12.11).
We therefore prove (12.11), and in doing this, there is no harm in
assuming that c
= 1 and Ys are all positive.
64
-,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
From the inequality
I Ys -11 < a ) I <- p( I Ys -11
s
s
6 being any given positive number, it follows that
I p(Xs
(12.13)
I
< aY )-P(X < aY ,
s
> 6),
sup [p{x ~ Xs < x')+p{ I xs-Il
x-x'i < 21 aI 6
P(Xs/Ys < a)-P{Xs < a) 1 <
I
> 6)
for all s.
Since [X } (s
s
exist, for any given
s
o
= 1,2, ••• )
e > 0,
has property B{S) by assumption, there
a positive number M and a positive integer
such that
Ixs I~ M)
p{
for all s ~ so.
< e/3 ,
Furthermore, property C{S) 'of (Xsl{s = 1,2, ••• )
assures us that there exist a constant 6
p{x < X < x') <
sup
I x-x1
for all s
<6
-
o
s
> 0 and a positive integer s'o such that
e/3
>
= s'.
0
Because of the il}Probability convergence of (Y } (s
s
= 1,2, ••• )
to 1,
there exists a positive integer s" such that
o
I Ys -11 >6 0 /(2M) ) <e/3,
p(
for all s
> s" •
=
0
Hence, putting 6
(12.14)
sup
-eo
for all s
= 6 o/(2M)
for 6 in (12.13), we have
s < a) - P(Xs < a) 1 < e
I P(X/Y
< a < ..
> max{s ,s',s") , which proves (12.11) and hence the theorem.
-
0
0
0
As an immediate consequence of this theorem, we have the following
COROLLARY 12.2. (a) Suppose that the conditi ons, (i) and (ii), of
the above theoremhold, and that [Xs}(s
= 1,.2, .. ,)
(X Hs = 1,2, ... ) as s ...... Then, it holds that
s
(12. 15 )
X /Y ,.. X / c (I, S) , ( s.....),
S
B
S
and
65
be ASEQ{I,S) to another
( 12.16)
y .... cX (I, S) , (s ....... ).
s s
s
If we replace the condition (i) of the above theorem by the
x
(b)
I
I
_I
condition
(1) Lx s }(s = 1,2, ••• ) converges in law to a certain distribution, Z,
of the c ::;ntinuous type as s .... eo
,
then it holds that
(12.17)
and
(12 .18)
xs /Ys
.... Z/c (I, ~ ) , (s...... ),
xs
.... cZ
Y
x
(I,S), (s ...... )
The proofs of these results are omitted.
Given a sequence of two-dimensional random variables l(xs'ys)}
(s
= 1,2, ••• ) with Ys .... c
question:
(in prob.) as s .... -, one may ask the follo',dng
What conditions on the sequence (Xs }(s
(I,S), (s ......),
Ys /X s .... c/Xs
=1,2, ••• )
guarantee that
_I
holds ?
This is still an unsolved question.
13.
PrOperties of type (I,S) aSYmPtotic independence.
The purpose of this section is to show some properties of type (I,S)
asymptotic independence, which are similar to that of type (I,m) asymptotic
independence given in the last of Section 9.
Let, as in Section 4, [x(n ) = (X~(m~)' ... 'X:s(~ »}(s = 1,2, ..• )
s
be a sequence of random variables, the system of whose mirginals
{
x~(~), ... ,X: (~~)}
s
s
are subjected for investigation, and let{ Y(n ) =
s
(s
(Y~(~)' ••• 'Y~s(~
»}
s
= 1,2 ••• ) be the corresponding sequence such that y~( m.s) is identically
~
distributed with x~(
s) for each i and s and the system
~ m.
~
66
I
I
I
I
I
I
~
I
I
I
I
I
I
I
_I
I
I
I
I_
(13.2)
is independent for each s.
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Now, 1et{ Z(n ) = (Z~(r\)' ... , Z:s ({
s
s
sequence of random variable, whose margina1s
(13.3)
{
Z~(
»} (s = 1,2, ... ) be another
8), .•• ,~ ( S )}
s ~s
~
is an independent system far each s.
Under this situation one
can prove
the following
If{ X(n )} (s = 1,2, ••. ) and{ Z(n )} (s = 1,2, .•• )
s
s
are ASEQ,(II,S) as s .. - and the-condition (1 +' ~I(x(n )'Z(n );S(n »)
k
s
s
s
s .. 1, (8 .. _) is satisfied, then the system (13.1) is ASIN(I,b) as s-+-.
THEOREM 13.1.
(a)
(b) If n are bounded uniformly for all 5, and [X(n »)(8 = 1,2, .•• )
s
_s
s
and lZ(n )J(s = 1,2, ••• ) areASEQ,(I,S) as .. - , then the system (13.1) is
s
ASIN(I,S) as s .. - •
PROOF.
The proof is quite similar to that of Theorem 9.3, as will
be seen below.
(a) We have to show that
X(n)"" Y(n)
st: .
~
(13.4)
I p (nS)(ECn »
S
for any sequence [ E(n»)
s
(I,S) as s.. -
S
- p (ns ) (E(n »
s
(s
= 1,2, ••• )
I..
, or equivalently
0, (s .. eo) ,
with E(n ) in S(n ) for each s.
s
s
Since every E(n ) is expressible in the form
s i s
s
E(n ) = El(mi)X•••X Eks(m: ) , Ei(m~) e S(m~)'
s
s
and
(13.5)
I
I
for each i and s, it is easy to see that
I p (nS)(E(n »
yS
ZS
(13.6)
- p (nS)(E
S
Cn s
»
I ~O,
_a
(S~IO),
from which (13.4) follows.
(b)
In this case, we need solely
I. pZ~(]. m.].s (Es
y~( s)
) p ]. m. (Es
i(m~) ].
i(m~)
].
].
instead of (13.5), the right-hand side of which tends to zero as
s~1O
,
by the assumption of the theorem.
Hence the proof of the theorem is completed.
It is not known yet whether the condition in (a) of the above
theorem can be weakened or not.
As an immediate consequence of the above theorem we have the
COROLLARY 13.1.
s
[X(n)
In the case of equal basic spaces, if a sequence
s ) } ( s = 1,2, ••• ) converges (
) to some
s
= (Xl(ml)'···'Sk(~)
I,S
distribution Zen)
= (Zl(~),···,Zk(~»' whose marginals [Zl(m )'···'
l
~(~)} being an independent system, then the system lX~(~), ••• ,~(~)l
is ASIN(I,S) as s .... , where k and m.]. 's are all independent of s.
The following result is also seen at once
For any given [X(n )
s
= (X~(m~)' ••• '~s(~
corresponding sequence tY(ns)}(s
»l(s • 1,2, ••• ), and the
s
= 1,2, ••• ) as was given
in
the beginning
of this section, let us put
(13.8)
XCm )
s
= f(n s ,ms )(x{ns »
and Y(m )
s
= fen s ,ms )(Yen s »
for each s, where f(n ,m )(Z(n » is a measurable transformation from
s s
s
R(n ) to R(m ).
s
Then we have
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
s
68
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
THEOREM ld.2.
Suppose that the transformtion preserves type (l,S)
asymptotic equivalence and is such that a system of marginals of Y(m ) ,
s
(~(tS , ..• ,~ (Js )} say, is independent for each s.
1)
s h
s
Then, the corresponding system of marginals of Xim )' t~(Js), ••• , ~ (tS )J,
s
1
s hs
is ASnf(l,S) as s.... - •
The proof of this theorem is omitted.
As an immediate consequence of this theorem we have the foUovling
COROLLARY 12.2.
Any subsystem of asymptotically independent (I,S)
system is asymptotically independent (1,5).
14.
Measurable transformations preserving tyPe (I, 5) asymptotic
equivalence in the general case.
Let lX s(
(s
= 1,2, ••• )
) = (X~, ••• ,Xs )} (s = 1,2, ••• ) and
ns
ns
( s
s
)= Yl'···'Yn
s
»
s
be two sequences of random variables with X(n )' Y(n )
s
s
belonging to ;,; (R(n ),<B(n » for each s.
s
s
(14.1)
LYs( n
f(ns,ms)(Z(n »
s
Let, further,
= (f(ns,l)l(Z(ns»),···,fCns,l)ms (Z(n s »)
be a measurable transformation from R(n ) to R(m ) for each s, where
e
s
n > m • These transformations define new sequences of induced random
s = s
variables, l~(m ) = (U~, ••• ,us )} (s
s
msl
(s = 1,2, ••• ):
= 1,2, ••• )
and (~(m )
s
= (~, •••
,v: )}
s
U(m ) = fen ,m )(x{n » and Vem ) = fen ,m )(Y(n » •
s
s s
s
s
s s
s
Clearly, these random variables belong to a (R(m ),CB(m
s
s
In this section we shall decide some of the classes of measurable
(14.2)
».
transformations which preserves type (1,5) asymptotic equivalence in the
general (equal and unequal) basic spaces.
In the first place, we shall consider a class of measurable
I
I
transformations of the form
_I
for each s.
For this type of transformations, we can prove the follOl'Ting
THEOREM 14.1. If, in (14.3),
f~(x)
is a continuous and monotone
~
non-decreasing function of x on the real line for each i and s, then it
holds that
s
s
Xen ) ~ Yen )(I,~),(s -+ 10) implies that U(n ) ~ V(n )
s
s
s
s
(1,.0, (s -+10), and
s
s
(b) X(n ) ~ Y(n )(I,S),(s-+1O ) implies that U(n ) ,.. V(n )
s
s
s
s
(I, S), (s-+ 10 ) •
(a)
PROOF.
If the function (14.3) satisfies the condition of the
theorem, then it holds that
-1
fCn
n) ().l(n ))
s' s
s
for each s, which yield
~
_I
Jo\(n )
s
and
~(X(ns )'Y(ns );S(ns )) ~
for each s.
dI(U(n )'v(n );S(n ))
s
s
s
These imply in turn (a) and (b) of the theorem.
As a special case of this theorem we have the following
COROLIARY 14. 1.
Suppose that the function (14.3) is of the form
s
(
s
s
s
s
fen n \ Z( )) = ( c l :3 l +d l ,
,c Z +d
)
s' s)
ns
·
n s ns n s
for each s, where c~ > 0 for all i and s and d~ are constants. Then, the
(14.4)
result (a) and (b) of the above theorem hold true.
It is not clear in general whether we can weaken the contitions
in the above theorem.
I
I
I
I
I
I
For a special case, however, where the distributions
involved are of the continuous type, we can see the following
70
I
I
I
I
I
I
I
_I
I
I
I
I_
's
•.8
THEOREM 14.2. If X(n ) and len ) are of the continuous type for
s
s
s
in
(14.3)
is
continuous
and monotone non-decreasing or
all s, and if f.(x)
~
non-increasing in x on the real line for each i and s, then
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
X(n )
s
N
Y{n ) (I,S) implies that U(n ) v(n ) (I,S) as s-. ••
s
s
s
The proof of this theorem is easy and omitted.
N
It should be noted that, in both of the above two theorems, if,
furthermore, the transformation (14.3) is from R(n ) onto itself and nons
singular (one-to-one and inverse transfornation is also measurable), then
the inverse conclusions of the results in both of the theorems are also true.
and
f(ns,ns)(Z(n » = (log zl,···,log Zn )' for Zi) 0,
s
s
preserve type (I,S) asymptotic equivalence.
(14.5)
EXAMPLE 14.2.
So far as we are concerned with sequences of random
variables of the ccntinuous type, the transformations
fen n )(Z(n » = (-zl'···'-z ),
s' s
s
ns
preserve type (I,S) asymptotic equivalence.
(14.6)
EXAMPLE 14.3.
Consider the transformations
fen n )(z(n » = (l/zl,···,l/z ), for 21> O.
s' s
s
ns
-log x
.
s
..s
Then, since 1/ x = e
, it is seen that ~f X(n ) and l(ri ) are of the
s
s
continuous type and their components are all positive for all s, then the
(14.7)
transformations defined by (14.7) preserve type (I,S) asymptotic equivalence of
lX(n )J(s
s
= 1,2, ••• )
and lY(n »)(s = 1,2, ••• ).
s
These examples come out from the preceding two theorems.
It is difficult to decide the classes of measurable transformations
which preserve type (I,S) asymptotic
equivalenc~,
'{l
except for the cases we
treated above.
However, we shall give a slight investigation to the case
_I
when the transformations are of the form
(14.8)
f(n ,m)(z(n »
s
s
= (f~(Z(n », ..• ,f:(Z(n ») ,
s
s
where m is fixed independently of s and m < n far at least sufficient ly
= s
large value s of s.
s
In this case, corresponding to the original sequences, [X(n )}
(s
= 1,2, ••• )
s
and [Yen )}(s
= 1,2, ••• ),
we have those of new random
s
variables of a fixed dimension, i.e., [U(m)}(s
= 1,2, ••• )
s
and lv(m~(s
= 1,2, ••• ).
Then, type (I,S) asymptotic equivalence is equivalent to type (I, ~ for
s
'
sequences (U(m)J(s
= 1,2, ••• )
s
}
and [V(m) (s
= 1,2, ••• ).
I
I
Since any subset
belonging to K(m) is open and the inverse image of any open set with respect
to a continuous transformation is open, we have the following
THEOREM 14.3. If the transformation (14.8) is continuous for
each s, then type (I, q) asymptotic equivalence of (X(ns »)( s
= 1,2, ••• ) and
I
I
I
I
I
I
_I
[Y(n )}(s = 1,2, ••• ) implies type (I,S) asymptotic equivalence of iU(m)J
s
(s
= 1,2, ••• ) and
lv(m)l
(s
= 1,2, ••• ).
Thus, if one can find conditions under which (I,Q) and (I,S) are
mutually equivalent, then under such conditions continuous transformations
of the form (14.8) preserve type (I,0) asymptotic equivalence.
It is thus
an interesting question to find out conditions under which two notions of
asymptotic equivalence, (I,q) and (I,S), are mutually equivalent.
There is another question in connection with type (I,S) asymptotic
equivalence preserving transformations:
Let us consider a situation, in
which n s increases monotonically to infinity with increasing s, and (X(n )}
s
(s = 1,2, ••• ) and lY(n )J (s = 1,2, ••• ) themselves are not ASEQ(I,S) as
s
s ~~, but any marginals of finite dimension, [XC ) = (X~ , ••• ,X~ )}
n
(s
= 1,2, ••• )
and
lY
s( )J
n
= (Y~
~l
, ••• , ~ ) (s
~n
72
= 1,2, .•. )
~l
~n
are ASEQ(I,b) as
I
I
I
I
I
I
I
_I
I
I
I
I_
s .... - for any given positive integer n and any given choice of i l ,." in both
independently of s. (This is equivalent to the follo',·.1.ng statement: Far
I
I
I
I
I
I
any given positive integer n, (X{n)
Ie
I
I
I
I
I
I
I
Ie
I
(y~, ..• ,~)}(s
= 1,2, ••• )
= (~,
,X:»)(s
are ASEQ(l,S) as s -.)
= 1,2, ••• )
and (Y(n)
=
One may call this type
of asymptotic equivalence of (x{n )l(s = 1,2, .• ) and [Yen »)(s = 1,2, ••• )
s
s
the finite aSYmPtotic equivalence {l,S) and say that these sequences are
finitely asymptotically equivalent (l,S) as s.... - •
NOvl,
we take measurable transformations which are of the form
(14.8), and suppose that there exists a sequence of sets of real numbers
lc~l(i
= l, ••• ,m; s = 1,2, ••• )
VS)}(s
m
= 1,2, ••• )
such that the sequence (V(m)
has properties
C(S) and B(S).
= (c~
~, ••• ,c:
The question is then to
find out conditions on functions (14.8) under which the finite asymptotic
equivalence of (X(n )}(S = 1,2, ••• ) and [Y(n )}(a = 1,2, ••• ) implies the
s
. s
S
• .B
type (l,S) asymptotic equivalence of [u(m)l(s = 1,2, ••• ) and lV(m)}(s = 1,2, ••• ).
The following example relates to this question.
EXAMPLE 14.4.
problem:
Let lX(n )l(s
The present author once met with the follmTing
= 1,2, ••• )
be a sequence of real random variables
with n s'1'eo as s .... : , such that (XCn)
= (X~, ••• ,X:)}(s = 1,2, ••• )
converges
in law to n-dimensional independent normal distribution with mean vector 0
and variance-covariance matrix I n (unit matriX) as s .... - , where n is any
positive integer fixed independently of s. Then, for any fixed n, the
n
variable 1:
(X~)2 converges in law to the chi-square distribution of
i=l
deSfiees of freedom n. Now, tle question is to ask whether the variable
s
1: (X~)2 is asymptotically approximated by the chi-square distribution
i=l J.
of degrees of freedom ns as s.... - , in the sense of type (l,S), vlhich is
still an open question.
73
15.
Measurable transformations preserving type (I,S) asymptotic equivalence
in the case of equal basic spaces.
I
I
_I
In this section we shall confine ourselves to the case of equal
basic spaces where n n for all s, and consider the same type of problems
s
as in the preceding section.
Suppose we are given two sequences of random variables, tX(n)}
(s
= 1,2, ••• )
and LY(n)}(s
= 1,2 ••• ),
and let us consider measurable
transformations from R(n) to R(m) (m ~ n), which are of the form
(15.1)
for s
= (f(n,l)(Z(m»,···,f(n,l)m(Z(m»'
f(n,n)(Z(n»
= 1,2, ••••
Firstly, we shall prepare some necessary lemmas.
Let
~(n)
be the class of all subsets of R(n) which are of the form
z. = b.
~k
b.
:5
~k
Zj
k
<
= 1, ••• , t
, and b
or
<z j <
eo ,
Jm m
m = 1, ••• , n- t
where t is any integer such that 1
-eo
~
t
m
~
.:::: Zj < a. ,
j m}
m
Jm
Zj , or -eo
<
m
n, lil, ••• ,i t }
or
'7
'~4n
<
_I
eo
is any subsequence
of ll, ••• ,n} with complementary subsequence ll, ••• ,n} with complementary
subsequence [jl, ••• ,j
.}, and a., b., i • l, ••• ,n, are real numbers.
n-h
~
~
Clearly, any member of this class has J.1(n)-measure zero.
* be the class of all n-dimensional intervals,
Furthermore, let Sen)
open, semi-open and closed.
Then, it is evident that this class contains
Sen) as a subclass, and that for any member E(n) of S *(n) there can be
found a member F(n) of Sen) such that the symmetric difference
E(n)
~
F(n)
is expressed as a union of at most 2n mutually disjoint members of ;r (n)'
He now prove the following
LEI;J!·1A 15. 1.
If (X(n))( s
ASEQ(I,S) as s-.+eo , then they are
= 1, 2, ••• )
ASEQ(I,~)
and (Y(n )} (s
as s
-.+eo •
= 1, 2, ••• )
I
I
I
I
I
I
are
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
PROOF.
as s ... -.
let E(n) be any given set belonging to 3' (n) ~'ith the form (15.2),
and E(~) be the subset obtained from E(n) by changing the first J conditions
in (15.2) into J, conditions, bi. ~ zi
Ie
I
K
_p
<
lip ; k = 1, .•• , J, where p = 1,2 •••.
k
P
Then, evidently, .t.(n) are members of Sen)
end E(n)4,. E(n) as p ... - •
Hence, by the condition of the theorem, there exists a sequence
of positive numbers, [q } (s = 1,2, ... ), with q ... 0 as s "'-, such that
s
s
Ipx(n)(E1n»
- pY(n)(E1n»
I< q. ' for
s1l P and .,
where qs are independent of p and E(n).
Ie
I
I
I
I
I
I
I
One has to show that
letting p tend to infinity, we then get
(15.4)
I
P
X(n)
(E(n»
Y(n»
- P
(E(n)
I
. ~ qs'
for all s, which is equivalent to ( i5 .3 ), and the proof of the lemma. is
completed.
By this lelllll&, it is inmediate that
LEMMA 15.2.
* and (l,S),
Two types of asymptotic equivalence, (l,S),
are mutually equivalent.
PROOF.
This follows from the inequality
~(X(n)'Y(n);S(n» ~
dl(X(n)'Y(n);S(n»
+
2n~(X(n)'Y(n);3'(n»'
for each s.
Now we shall consider the case where the transformations (15.1)
are specially of the form
and put
(15.6)
75
I
I
Then one can see easily that
THEOREM 15. 1.
Suppose that the i-th component of (15.5) , f~(x),
J.
_I
is a monotone (non-decreasing or non-increasing) function of x on the real
line for each i and s.
(I,O)
Then XCn ) - Y(n) (I,S) implies that U(n) - v(n)
-.~.
as s
The proof of this theorem is easy and is omitted.
As immediate consequences of this theorem we have the following
= 1,2, ••• ) converges (I,S) to some Y(n)'
COROLLARY 15.1. If tX(n)}(s
or converges in law to some Y(n) of the continuous type, then, under the
condition of the theorem, it holds that U(n) - v(n) (I,S) as s-.~ , where v(n)
is given by changing YCn) into Y(n) in the second definition of (15.6).
Furthermore, if the transformation (15.5) is independent of s, then under
the above conditions it holds that U(n) -,V(n) (I,Ll) as
COROLLARY 15.2.
S-,OQ.
Suppose that the function (15.5)
t~ces
a special
I
I
I
I
I
I
_I
form:
s
(
)
f(n,n) z(n)
( 15.7)
= (s
c l zl
s
where c~ and d~ are any real numbers.
implies that U(n) - v(n)
s
s)
+ d1,···,c n zn + dn
'
Then X(n) .... Yen) (I,S)
(I, S) as s -'10 •
The proofs of these corollaries are omitted.
We shall state another special case in the following
E~·~LE
(15.8)
15.1.
Let the function (15.5) be given by
f( n,n )(z( n »=(l/Z 1,1/'1.n )for all z.J.
f
0 , and arbitrary when z.J.
for some i.
Then, it is easy to see that, if P(X~
X(n) - Y(n)
(I,S) implies that U(n) - v(n) (I,S) as s-.· •
J.
= 0) = 0
=0
for each i and s,
Evidently, the above result follows from the fact that the
function l/x consists of tyro monotonic parts, though the function itself
is not monotone as a whole.
Similar argument can be applied to get the following
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
THEOREM 15.2.
Suppose that the i-th component of the function
(15.8), f~(x , consists of k: monotonic parts for each i and s, and k~ ~ K
uniformly for all i and s.
Then the transformations preserve type (I,S)
asymptotic equivalence.
The proof of this theorem is easy by using Lemna 15.2, and is
omitted.
If f~(X) in (15.5) is a polinomial in x of degree
EXAMPLE 15.2.
k~ for each s and i, and k~ are bounded uniformly for all i and s, then
(15.5) preserve type (I,S) asymptotic equivalence.
Nm'1, in
the next place, we shall consider continuous transformations.
Let us consider a measurable transformation
and let
U(m)
= f(n,m)(x(n»
and v(m)
= f(n,m)(Y(n»'
and, in a convergence case where Y(n) are identical with some Y(n)' V(m)
=hn. tn/Y:1I').
A classical result states that
LEMMA 15.3.
Suppose that the function (15.9) is continuous, Le.,
every component of the function is continuous in Z(n) over R(n).
law convergence of (X(n») (s
= 1,2, ••• ) to Y(n) implies the same type of
= 1,2, ••• )
convergence of (uf,(n )1 (s
Then, in
to Y( n ) as s .... - .
A version of this result in our case is given by the following
THEOREM 15.3.
Suppose that the function (15.9) is continuous and
one of the sequences, (X(n)} (s
properties c(S) and B(S).
«I,S»
as s .... - .
= 1,2, ••• )
and tY(n)l
(s
= 1,2, ••. ),
has
Then, X(n) - Y(n) «I,MO) implies U(m)- V(m)
If, moreover, the function transfers
C(S) and B(S) of the original sequences, then X(n) - Y(n)
U(m)'" VCm) (I,S) as s .... - •
77
the properties
(I,~) implies
PROOF.
First, it is noted that, under the conditions in the first
part of the theorem, type «I,~) asymptotic equivalence of tX(n)} (s
and tY(n») (s
= 1,2, ••• ) is equivalent to type
= 1,2, ••• )
(I,~) asymptotic equivalence
of the same sequences, and under the condition in the second part of the
theorem, «I,~»
and{V(m)}(S
and (I,S) are mutually equivalent for [U(m)}
(s
= 1,2, •.• )
= 1,2, ••• ). Hence the second part follows immediately from the
first part.
We shall show the first part.
To prove this, it suffices to show
that
(15.10 )
Since the inverse image of any member of )'(m) with respect to the
function (15.9) is open and hence belongs to q(n) , (15.10) follows easily
from Theorem 7.4, or Corollary 7.2, and the conditions of this theorem.
I
I
_I
I
I
I
I
I
I
_I
This proves the theorem.
From this theorem, it is immediate that
COROLLARY 15.3.
convergence of tX(n)} (s
If the function (15.9) is continuous, then in law
= 1,2, ••• )
to same distribution Y(n) of the
continuous type implies type (I,S) convergence of [U(m»)
(s
= 1,2, ••• )
to V(m) •
Some of those transformations Which transfer the properties C(S )
and B(S) simulteneously are shown in the following
EXM·1PLE 15.3.
The fo llm'ling transformati ons are easi ly seen to
transfer the described properties: f(x) = e X,( __ < x <_ ); f(x) = e- x ,
(--< x < 10); f(x)
f(x)
= log(l
= xk
, (--< x <eo), for any given positive integer k
+ x), ( 0 < x ~ ),
= 0,
It should be noted that if
(otherwise).
\'Te
take transformati ons depending on s
instead of (15.9), the problem become quite difficult in general.
If,
however,ie could find suitable conditions under which type (I,S) and type
78
I
I
I
I
I
I
I
_I
I
I
I
I_
(I,Q) asymptotic equivalence are mutually equivalent, it will be possible
to clearify the type (I,S) asymptotic equivalence preserving property of
the transformations, which is left to future investigations.
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Now, in the final place, we shall consider the following type
af
transformations.
s
n s
n
2
( 15 11)
f
( z ) = (. I: c. zi' I: d S z )
.
(n,2) (n)
i=l ~
i=l i i '
where c~ and d~ ( > 0) are constants for all s and 1, 1=1, ... ,n; s = 1,2,
s
s
(_.8)
Let, further, U(2) = f(n,2) X(n)
S
S
(s
and V(2) = f(n,2) Yen) for all s.
Then, we can show the following
THEOREM 15.4. Under the s1tuation s.tated above., if one of the
sequences, (X(n)} (s = 1,2, ... ) and {Yen)} (s = 1,2, •.• ), has properties
C(S) and B(S), then X(n)
PROOF.
s
IV
Y(n) (I,l:)
implies U(2)
IV
12)
(I,S) as S-+-.
Firstly, it is noted that there is no harm in assuming
s
that c and d are bounded uniformly for all i and s, because type (I,S)
i
i
asymptotic equivalence of UsC ) and ~( ) is equivalent, by Corollary 15 2,
n
n
n
n n
n
.8 , I: d.
-s( Y.S)2) , ,-There
-s s, I: d-s( XiS)2) and (-S.
to that of
(I: c.X
I:
CiY
i
i
i
i=l ~
i=l
i=l
i=l ~ ~
n
c~=c~/p and d~=d~/a
~
~
/
n
/
with p =(I: (c~)2)1 2 and q = ( I: (d~)2)1 2 .
s
~
~ ~
s 1=1 ~
s
i=l ~
s
Let us take any convergent subsequences of (cls,
... ,cn )} (8 = 1,2,
.
and [d~, ••• ,d:)} (s = 1,2, ••• ) , and without any loss of generality we
suppose the original sequences are convergent, i.e.,
s
s
c. -+ c
and di -+ di
' as s -+ - ,
~
i
for each i and for some c. and d. ( > 0).
~
Let
~
=
s
a and b( > 0) be any given real numbers, and let A(n)
n
8
2
I: d. z. < b} ,
i=l ~ ~
79
and
. )
for each s.
Then, it is evident that the set A(n)nB(n) is the inverse
image of the set
E(2)
= (Z(2) I
;1 < a , ~2 < bJ with respect to the
I
I
_I
function (15.11).
~t
us put
n
A(n)
= (Z(n) I i~l
c i zi < a
and B(n)
= L z(n) I
n
2
i~l di z i > bJ .
Then, the sets A(n) and B(n) coincide in the limit as s ~. with A(n) and B(n)
s
up to their boundry sets, respectively.
s
Hence A(n) nB(n) coincides with
A(n)n B(n) up to their boundary set in the limit as
S~.
By the assumption of the theorem for any given P > 0, there exist a
positive integer So and a member M(n) of S(n) , whose closure being compact,
such that
el
for all s > s •
==
0
Now, put
s
K
for each s.
BS M
(n)n (n)(\ (n)
- AS
(n) -
an
d K
= A(n)n B(n)n
(n)
M
(n),
Then, K(n) coincides with K(n) up to their boundary sets in
the limit as s
~eo ,
and the closure of K(n) is compact"
Note that K(n)
s
and K(n) are all dependent on the values of a and b we have given.
Now, it is seen that, for any given positive 6 , we can find a partition
of M(n) consisting of the members of ;;)(n),(E(n)i} (i
such that
-
= 1, ..• ,
Nil) say,
.
sup lJ1(n)(Ej E(n)j) I E(n)jnK(n) 1= ¢} < 6
-o~~ :
where E and f denote in general the closure and the boundary set of the set E
(15.13)
I
I
I
I
I
I
For any given such finite partition, the supremum in (15.13) is
attained for some values of a and b, a
o
and b
0
say, because there exists
I
I
I
I
I
I
I
_I
I
I
I
I_
positive number h such that h < a and h < b imply that K(n) = M(n)' and
I
I
I
I
I
I
is an inner point of the set
For all values of a and b, any point of K(n)
r
E(n)j' unless it lies on the bQundary of M(n).
Giving a finite partition of M(n) satisfying the condition (15.13), let,
for each S,[E(n)~} (k = 1, ••• ,N~) be the set of all such members of the
partition that E(n)ikn K{n) r~, an~ among these, let [E(n)~) (k = 1, .•• ,Ns )
be the set of all such that E(n)LnK
= ~ • Clearly the choice of til' ... '~'}
~ (n)
s
is dependent on a, b and 8 . Then, we have
~
rl
I p (n)(Ks )-p (n)(Ks
.)
(n)
(n)
(15.14)
1<1
=
~
rl
N
p (N)(Es E
)-p
.k=l (n)~
(n)(~ E-
)1
.
k=l:~o)~
xen) (EN;
Y(n) Nt
E(n)i_ )+P
(E E(n)i )
k=N +1 ~
k=N +1 k
+P
B
for each s.
Ie
I
I
I
I
I
I
I
Ie
I
•
a < -h implies that K(n) =~.
S
Choosing 6 in (15.13) sufficiently small and a corresponding partition
of M(n)' it follows from the_property C(S) of the original s~quences that
~
s
~
N'-
lini l P tn) (E Ern)i ) < eo
S
-+l
k=N +1
s
and
k
N'
.
lim p (n) (E S E( )i ) < e
S-+k=N· +1 n ~
s
,
uniformly for all values of a and b.
s ~ Nil, the condition X(n)- Y(n) (1,5) as
-+ -implies that the first term on the right-hand side of (15.14) tends to
On the other hand, since N
S
zero as s
-+- .
Hence it follows from (15.14) that
X
(15.15)
sl~m_lp
~
S
(n)(K(n»
- P
I
(n~(K(n)L ~ 2
t
,
from which, by (15.12), we have
(15.16)
s~~
l
X
S
p (n)(A(n)nB(n»
uniformly for all a and b.
~
- P (n)(A(n)nB(n»I
~
4
e ,
(V(2)}
This means that (U(2)} (s = 1,2, ••• ) and
asymptotically equivalent in the sense of type
(I,~,
(s = 1,2 ••• ) are
and hence in the
I
I
_I
sense of type (I,S) as s -t. , which proves the theorem.
The following result is immediate from this theorem.
COROLLARY 15.4.
Let lX(n)} (s = 1,2, .•• ) be a sequence of random
variables which converges in law to a certain distribution Yen) of the
continuous type as s -t. , and let U(2) = f(n,2)(X Cn » and
(Y (n».
Then, U(2)'"
V(2)
V(2) = f(n,2)
(I, S) as S-t·.
Taking marginals of U(2) and v(2) we have
COROLLARY 15.5.
theorem, X(n)'" Y(n)
n
(15 •15 )
as s
s
Under the same condition as that of the above
(I,S) implies that
s
Z coX
i=l ~ ~
0
n
...
s
s
(I, s)
Z coY
isl ~ ~
0
-t • •
COROLLARY 15.6.
If (X(n)}
which is of the continuous type as
(15.16)
n
s
s
~
~
n
(s
= 1,2, ••• )
S-t 10
,
converges in law to Yen)
then it holds that
s
Yo
(I,S), (s -t.).
i
~
In the last place, we shall consider the following example.
Z co Xo ... Z c
o
~=
1
0
~=
1
) = (x:, ••• ,X~)j (s = 1,2, ••• ) be a sequence
Cn
of random variables which converges in law to the n-dimensional independent
EXAMPLE 15.5.
Let tX
normal distribution with mean vector 0 and variance-covariance matrix I ,
n
and put
n
n
s 2
V = ~ (Yo + co)
~
s i=l ~
s
for all s, co~ are any gi ven constants. Clearly, Vs is distributed according
=~
(XiS + c S )2
l
sO
l
~=
U
and
to a non-central chi-square distribution of degrees of freedom n ,nth non2
n
s 2
centrality parameter p = ~ (co) , for each s.
s i=l ~
For (15.17) we can see that
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
us -
(15.18)
In
V
s
(I,S), (s....).
fact, these variable s can be rewritten as
n
s 2
n B..S
2
U = ~ (Xi) + 2 ~ cixi + P
s i-l
i=l
s
V
s
for each s.
n_.2
= E Y1
i=l
n s
+ 2 E c.Y.
i=l
+
~ ~
2
Ps
Then, by Corollary 14.1 or Corollary 15.2, one can see that
the condition (15.18) is equivalent to the following
n •.82
ss n_-2
n s
(15.20) E (xi) + 2Ec i Xi - E "Ii + 2E ciYi (I,S), (s....).
i=l
i=l
i=l
i=l
Now, CorollarY 15.4 assures us that
n ..8 2 n s s
n _.2 n 8..
(E (Xi) ,E ciXi )-( E Y1'~ Cil i ) (I,5), (s.... ).
i=l n
inl
i=l i=l
But, since (E
E c~ Yi )} (8 = 1,2, ••• )18 = 1,2, •.. ) has properties C(S)
i=l i=l
and B(5)(unless c~ are all zero for i=l, ••• ,n and for an infinite number of
ti,
values of s), (15.20) follows from Corollary 15.5.
16.
Type (I,5) Asymptotic equivalence of marginal random variables.
General formulation of the problem we want to treat in this section is
as follows:
Let (X(k)'y(J)'Z(m»J(s = 1,2, ••• ) be a sequence of n (= k + l + m)
dimensional random variables with fixed k,
~/
(i) for the first marginals, (x{k)}(s
se~.ence of k-dimensional random variables,
and m.
Far this sequence, suppose that
= 1,2, ••• ), there is another
(x(J)}(s = 1,2, ••• ), which is ASEQ
(I, S) to the original sequence as s.... , 8l1d
= 1,2, ••• ), there can be
found a sequence of real vectors (c;=(c~, ••• ,c;)} (8 = 1,2, ••• ), such that
the sequence (~/c~, ••• ,~/c;)}(s = 1,2, ••• ) converges in probability to
(ii) for the second marginals, (Y(J)J (s
the point (1, ••• ,1).
Under this situation, suppose that we replace the first two marginals
S
• .8
S)::S
S
of ( X(k)' 1(-4 )'Z(m) by X(k) 8l1d c(J) respectively for each s,
83
•
keep~ng
the
dependence structure between three marginals of the original variable undes
stroyed in a sense, to get the new variable (X{K),c(I)'Z(m».
I
I
_I
The problem is now to ask "lhether the third marginals of them,
LZ(m)}(s
= 1,2, ••• )
= 1,2, ••• )
and Lzem)} (s
are ASEQ(I,S) as s-+~, probably
under some additional conditions.
In the first place, some necessary lemmas are stated.
The following two lemmas are easy to verify and the proofs will be
omitted.
LEMMA 16.1.
Let tUs(X(n»}(s
= 1,2, ••• ) be a sequence of real valued
functions defined over R(n) , and suppose that the following conditions are
satisfied:
(i) For each s, 0 ~ us(X(n»
(ii) For each s, us(X(n»
~ for all x(n) in R(n).
is continuous over R(n).
_I
Then, there exist a convergent subsequence tUs'(X(n») (sJ+~) and a continuous func'Gion Uo (X(n»
such that 0 ~ Uo (X(n»
~ lover R(n) and us' (X(n»
as s '-+ eo for all x(n) in R(n)"
LE1-lMA 16.2.
The convergence us' (x(n»
(s-+~) in the above
-+ Uo (x(n»
lemma. is uniform on any given compact subset of R(n)' that is, for any
compact subset E(n) of R(n) and any given e > 0, there exists a positive
integer so' depending only on
e
and E(n)' such that
< e for all X(n) in E(n) and for all s'
Now, let f(X{n)'y (m»
~
I uS'(X(n»
- uo(X(n»
so.
be a real valued function defined over
R(n) x R(m)' the (n + m)-dimensional euclidean space.
Then, it is easy to
see the following
LEMMA 16.3.
Suppose that the function f(X(n),y(m»
with respect to (X(n),y(m»
over R{n+m).
R(n)' whose closure is compact.
is continuous
Let E(n) be any given subset of
Then, for any given point y(m) in R(m)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
_I
I
I
I
I_
and any given P- >0, there exilts a positive number 6 , depending on t, E(n)
I
I
I
I
I
I
< P. uniformly for all X(n)' in R(n)·
Ie
I
I
I
I
I
I
I
Ie
I
and Y(m)' such that I Y(m)-Y(m)I < 6 implies that I f(X(.)'Y(m»
PROOF.
(16.1)
- f(X(n)'Y(m)I
Put
v(Y(m)'Y(m»
=X
sUP I f(X(n)'Y(m»-f(X(n),Y'(m»I
(n) E E (n)
Suppose now that the assertion of the lemma is false. T~n, there
o
exist a positive number P, a point Y(m) in R(m) and a sequence of points
i
i
0
(Y(m)} (i = 1,2, ..• ) in R(m) such that Y(m) -+Y(m) (t -+.) ~d
(16.2)
~
v(Y(m)'Y(m»
p , i = 1,2, ••.
Since E(n)' the closure of E(n) , is compact by assumption, there exists,
o
for any given Y(m) and Y(m) , a point xfn) in E(n)' depending probably on
Y(m) and Y(m)' such that
= I f(X(n)'Y(m»
v(Y(m)'Y(m»
- f(X(n)'Y(m»I .
Then, corresponding to the sequence (X(m)} (i
oil
i
0
v(Y(m)'Y(m» = f(X(n)'Y(m»
(16.3)
for each 1.
= 1,2, ... )
in E(n) such that
i
i)1
- f(x(n)'Y(m)
i'
This sequence has a convergent subsequence (x(n)} (i '-+. )
whose limit point 1'n) being, of course, in E(n) .
Since f(X(n)'Y(m»
i'
0
(X(n)'Y(m»
*
0
-+(X(n)'Y(m»
i
i'
is continuous, (X(n)'Y(m»
*
0
-+(X(n)'Y(m»
and
as i'-+ .. , we have
i'
i'
* 0 I
I f(X(n)'Y(m»-f(X(n)'Y(m»
I
-+ 0, (:1: -+.),
and
Hence
o
i'
v(Y(m)'Y(m»
=
i'
0
i'
i' I
I f(X(n)'Y(m»-f(X(n)'Y(m»
-+
which contradicts (16.2).
This proves the lemma.
85
I
0, (i -+.),
I
I
Using this lemma one can show the following
IEMMA 16.4.
Suppose tha.t a function f(X(n)'Y (m»
_I
defined over
R(n) x R(m) satisfies the following conditions,
(i)
~ 0 over R(ntm)' and
f(X(n)'Y(m»
I
D( n ) = (X( n )
(ii)
K.
< x"i
=
~
~=
f(X(n)'Y(m»
(iii)
1, , .,k}
=0
over (R(n)-D(n»)X R(m) with
for some constants, X,
~
is continuous in (X(n)'Y(m»
For any given Y(m) in R(m)' f(x(n)'Y(m»
R(n) with respect to the ordinary Lebesgue measure
"K
n .
over the set D(n) x R(m)·
is integrable over
~(n)
,
and
J R(n)
d~(n) = 1
f(X(n)'Y(m»
,
for every Y(m) in R(m)'
Then, for any given zen)
(16.4)
u(z(n)'Y(m»
= JE
= (zl"",zn)
(n)
(z
(n)
in R(n)' the function of Y(m)
) f(X(n)'Y(m»
d~(n)
is continuous with respect to Y(m) over R(m)' where the set E(n)(Z(n»
is
given by
E(n)(z(n»
= [X(n) = (Xl'···'Xn )
I Xi
< zi ' i
= l, ... ,n},
and z,~ 's are allowable to be + .. for some or all i.
Firstly, let us consider the case when zi <
PROOF.
i.
In
to
for all
this case, we have, far any given zen)' Y(m) and Y(m) ,
I u(Z(n)'Y(m»-u(Z(n)'Y(m»)I
~i'
I f(x(n)'Y(m»-f(X(n)'Y(m»I d~(n)
J
<
D(n)nE(n)(Z(n»
.
suplf(X(n)'Y(m»-f(X(n)'Y(m»I~ (n)(A(n)nE(n)(Z(n»)'
where D(n) designates the same set as in the condition (i) and the 'sup'
is taken with respect to x(n) over the set D(n)n E(n)(Z(n»'
Since the
closure of this set is compact, Lemma 16.3 can be easily applied to get
the result.
86
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
Suppose next that some zi' s are ..... , ~+ 1= . zn=+ • say
In this
i
i '
i
case, putting zen) = (zl'·· "zk' Zk+l'···'Z~) with any sequence Sj ~ ..
(i ~.), j=k+l, •.. ,n, we have by the condition (iii)
u(Z(n)'Y(m»
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
But, since u(Z(n)'Y(m»
~
u(Z(n)'Y(m»' (i
~.
)
is a continuous function of Y(m) for each i as was
proved above, and since u(Z(n)'Y(m»
the limit function u(Z(n)'Y(m»
exists by (iii), we can conclude that
is a continuous function of Y(m) for any
fixed zen) in our case.
Finally, in the case when zi=..... for all i, the result follows from (iii)
This completes the proof of 1h e lemma.
Now, we shall return to the problem stated in the beginning of this
section.
In the first place, let us c,nsider the case when
t = O.
Let (X(n)'Z(m»}(s = 1,2, .•• ) be a sequence of (n + m)-dimensional
random variables, for which n and m are assumed to be fixed independently
of s.
Cumulative distribution functions of Z(m) and X(n) are denoted
respectively by Hs(z(m»
and Fs(x(n»' and the conditional cwnulative
distribution function of ZCm) given XCn ) = x(n) is denoted by Ps(z(m)lx(n»'
for each s. Clearly
(16.5)
Hs(Z(m»
=j R
Ps(z(m)lx(n» dFs(x(n»·
(n)
Let us ronsider another sequence of n-dimensional random variables
(X{n)(s = 1,2, ••• )
(16.6)
with cdf.'s
Hs(Z(m»
for each s.
Fs(X(n»' and let us put
=Ju
Ps(z(m) IX(n)
) dFs(X(n» ,
(n)
Then it is evident that this gives a cdf. of some n-dimensional
random variable, Z(m) say, for each s.
Under this situation, one can prove the following
THEOREM 16.1.
Suppose that the following conditions are satisfied:
87
(i)
I
I
For any given z(m) in R(m)' Ps(z(m)1 X(n)) is a c~ntinuous
_I
function of x(n) over R(n)
[X(n)}(s
(ii)
s
~
= 1,2, ... ) and CX(n)} (s = 1,2,
•• J are ASEQ,(I,S) as
••
(iii) There exist two sequences of real vectors, tC(n)
= (c~, ••• , c~)}
= 1,2, •.• ) and [d(n) = (d~, .•• ,d:)}(S = 1,2 ••• ), such that for one of the
(s
sequences in (ii) above, {X{n))(s
(U(n)
= 1,2, ••• )
say, the random variables.
= «~-d~)/c~, •.• ,(X:-d~)lC~)}(S = 1,2 ••• ).
converges (I,S) to some fixed distribution U(n) as s
~.
Then it holds that
(16.7)
ass~lO.
PROOF.
_I
If we put
s
U(n)
=
« Xl-d
s s)/ c s ,···, (~s
Xn-dns)/ cns) '
1
l
for each s, the condition (ii) of this theorem and Corollary 15.2 assure
us that
(16.8)
U(n)'" U(n) (I,S) as s ~., and hence
u(n) -+ U(n) (1,1:;)), (s -+10).
Hence, of course, both of the sequences [U(n)}(s ~ 1,2,
(s
= 1,2, ••• ) have property
B(b).
) and (U(n)}
Let us denote the cdf. 's of U(n)' uCn)
and U(n) by Ms(X(n))> Ms(X(n)) and M(X(n))' :respectively.
In order to show
(16.7), it suffices to prove that
sup
I Hs(Z(m)) - Hs(Z(m)) I -+ 0, (8 ~.),
Z(m) E: R(m)
because two t~es of asymptotic equivalence, (I,S) and (I,~), are mutually
(16.9)
equivalent in our present case.
Let e be any given positive number.
s
a(m)
s
s)
= ( al,···,a
m
Then, one can find a point
in R(m) , same of the components of which might be +
such that
88
I
I
I
I
I
I
10 ,
I
I
I
I
I
I
I
_I
I
I
I
I_
(16.10)
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(16.11)
sup
I Hs(Z(m»-Hs(Z(m»
z(m) E: R(m)
I < I Hs(&(m»-Hs(&(m» I + e .
It is seen, putting
s
u(n)
s
s
= ( cs
1 X1 +d!,···,cnx n
s )
+ dn
for x(n) = (x1 , .•• ,xn ), that
(16.12)
Ms (X(n» = F s (U(n») and Ms (X(n» = Fs (U(n» ,
It is also noted that the condition (iii) and (16.8) give us
for all s.
(16.14)
SUPR I Ms(X(n»
x(n)E: ,(n)
-M(X(n»\-+ 0, (s-+-).
.
Now, it follows from (16.5), (16.6) and (16.12) that
\ Hs(a(m»
(16.15)
=1
- Hs(a(m» \
JR
ps(a(m) \U(n»
dMs(X(n»
-j R
(n)
Ps(&(m) \U(n»
c1Ms (x(n»
I.
(n)
Since, for each s, Ps(a(m)lu(n»
is a function of x(n) , let us put this
function as
(16.16)
vs(x(n»
for each s.
= Ps(a(m)\U(n»
,
Then, by the condition (i) of the theorem, Vs(X(n»
is a
continuous function of x(n) over R(n) for each s, and
(16.17)
0 < v (X( » < 1
s
n -
uniformly for all s.
over R( )
n
Hence, by Lemmas 16.1 and 16.2, there exist a sub-
sequence (Vs1(X(n»)) (Sl-+_) and a limit function v(x(n»
(16.18)
such that
0 ~ V(X(n» ~ 1, and Vs1(X(n»-+ V(X(n»' (s-+- ).
This last convergence is uniform on any compact subset of R(n).
By
property B(S) of both sequences, ( U(n») (s
= 1,2, ... )
and
(u(n)}(s = 1,2, ••• ), there can be found a member B(n) of Sen)' whose
closure being compact, such that
89
usC)
p n (B(n»
(16.19)
for all s > s
-
USC)
> 1-e , P n (B (n) ) > 1-e
u()
n (B (n) ) > 1~
and p
for some positive integer s , where
0
.
0
e
I
I
,
is the same as that
of (16.10).
From (16.17) and (16.19), it then follows that
IJR
(16.20)
vs(x(n»
-J B
dMs(x(n»
(n)
vs(x(n»
I JR
vs (x(n»
-j
dMs(x(n»
B
(n)
vs(x(n»
dMs(x(n»
I
< e.
,
vs (X(n»
dM(X(n) )
I
<e
,
(n)
-J
dM(x(n»
B
(n)
=
dMs(X(n»I
(n)
I JR
for all s > s
<e ,
vs(x(n»
(n)
•
0
Since, as was stated above, vs,(x(n»
converges to V(X(n»
uniformly
integer s'0 such that
over B( n ) , there exists a pOSitive
.
JB
I
(16.21)
Ij B
V
s t (X(n»
(n)
vs(X(n»
dMs,(x(n»
-
(n)
dMs ' (X(n»
<e ,
el
j
dMst(x(n»
<e ,
I
I
I
I
I
I
I
B v(x(n»
(n)
for all st > st •
=
0
On the other hand, by (16.1.3) and (16.14), the Helly-Bray theorem
can be used to show that there exists a positive integer
IJ B
(16.22)
S"
o
dMs (X(n»
-j
<e
I J B(n) V(X(n»
elMs (X(n»
-j
<
,
(n)
for all s > s" •
0
It now follows
, from (16.20), (16.21) and (16.22) that
(16.23)
such that
V(X(n»
=
S)
I Hst(a(lll)
st
- Hst(a(m»
or equivalently, by (16.10),
I
I
I
I
I
I
I
V(X(n»
B
, (n)
-J
dMs t (X(n»
_I
< 6e ,
e ,
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
(16.24)
sup
I Hs,(z(m»
Z(m) E R(m)
- Hs,(z(m»
1< 7 f',
for a.ll s' >
max(s 0' s'0' s")
•
=
0
The above argument tells us that, i f (16.9) is not true, then we always
have a contradiction, which completes the proof of the theorem.
It should be noted that the last condition, (iii), of this theorem
can be replaced by tre following
(1ii)* U(n) converges in law to same distribution U(n) of the
continuous type.
s
Under the same situation as above, suppose that Z(m) has a conditional
s
.
probability density function given X(n) = x(n) , ps(z(m)IX(n»' and hence the
pdf. of the marginal Zem) is given by
(16.25)
hs(Z(m»
=J
ps(z(m)lx(n»
dFs(x(n»
,
R(n)
for each s.
For the sequence (X{n)} (s
(16.26)
hs(Z(m»
=
J
= 1,2, .•• ),
ps(z(m)1 x(n»
put
dFs(X(n»
,
R(n)
for each s, and let (Z(m)} (8 = 1,2, ••• ) be the corresponding sequence.
~mma
Then, by using
COROLLARY 16.1.
16.4, it is immediate from the above theorem that
Suppose that the conditions (ii) and (iii) of the
preceding theorem and the following condition are satisfied.
(i') For each s, ps(Z(m)lx(n»
(Z(m)'X(n»
is continuous with respect to
over R(m) x D(n)' with D(n)
set of constants Ki
IS,
= (X(n)IKi ~
and ps(z(m)IX(n»
=0
xi.,i=l.,n}
for some
over R(m) x (R(n)-D(n»' for all s
Then, it holds that Zem) /'V Z(m) (I,S), as s .....
The proof of this result is omitted.
In the second place, we shall consider the case when n = 0 in the
general formulation of the problem stated in the beginning of this section.
91
Let (Y(l),ZCm») (8
for which
.t
= 1,2, ••• ) be a sequence of random variables,
and m are fixed independently of s
Further, leU(c(l)
=
I
I
_I
(c~, ••• ,cs») (s = 1,2, ••• ) be a sequence of vectors Whose components are
all positive, for uhich it is assumed that, for each s, the point Cel) lies
in the defining subset Del) of
Y(l)' i.e., the subset of R(l) on which Y(l)
is defined.
Let Hs(Z(m»' Ps(Z(m)IY(l»
cdf. of Zem) given Yel)
= Y(l)
(16.27)
-J
Hs(Z(m»
and Gs(Y(l)
be cdf. of Zem) , conditional
and cdf. of Y(l) , respectively.
Ps(z(m) Y(I»
Then clearly
dGs(Y(l)
R(J)
Let us now put
(16.28)
s
Then this is a cdf. of some m-dimensional distribution, ~(m) say.
Under this situation, one can show the following
THEOREM 16.2. Suppose that the following conditions are satisfied:
(i)
For each s, and for any given z(m) in R(m)' the conditional cdf.
Ps(Z(m)IY(l»
(ii)
(16.29)
is a continuous function of Y(l) over
R(l)
The sequence of l-dimensional random variable given by
Y(~)
= (Y~/c~, ... ,~/c~) , s = 1,2, ... ,
converges in probability to the point 1($) = (1, •.• ,1) as s -+-.
Then, it holds that
(16.;0 )
Zem) -
zem)
(I,s)
as s -+ - •
PROOF.
Put, forY(l)
s
= (Yl'''',Y,),
( s
s)
vel) = clYl""'c Y
.
Then the cdt. of Y(l) of (16.29) is given by
Gs(Y(l»
= Gs(v(l»
,
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
and hence the cd!. of Z(m) given by (16.27) is expressed as
(16.31)
Hs(Z(m»
=j
Ps(z(m)
Iv(~»
R(I)
For any given e > 0, there exists a
sup R I Hi~Z(m»-Hs(Z(m»
Z(m) E (m)
far each s.
I<
(16.32)
dOs(Y(J»
poL~t
IHs(a(m»-Hs(a(m»
Now, let 6 be an;y given positive number.
I Hs(a(m»
IJ
=
(16.33)
~
+
'I
Y(l)-l(.t) ~ 6
J
I+e,
Then, it holds that
- HS(a(m»I
R
(ps(a(m) IvC.e»-ps(a(m)
(.e)
jI
s
a(m) in R(m) such that
I'
I> 6
Y(A) -1(,t)
Ic(~»)
dGs(Y(J»
I
I uS(Y(J» I dOs(Y(,t)
I uS(Y(A»
/ dOs(Y(J»
,
where we have put
(16.34)
us(Y(l»
= ps(a(m)IY(J»
- ps(a(m)/C(J»
Evidently, vs(Yce)iS a continuous and bounded function of Y(.e) over R(J),
and Us(l(J»
By
= O.
Lemmas 16.1 and 16.2, there exist a subsequence
(s' - 1,2, ••• ) and a continuous function Uo(Y(J»
Uo(Y(t»
t
us' (Yet»J
such that Us'(Y(t»
~
as s~ .. ; this convergence is uniform for all Yet) such that
I Y<.e)-l(.t)I~
0
and Uo(\;t»
= 1.
It follows then that there exists a
positive integer ssuch that
o
(16.35)
I
sup
Y (J) -1(.t)11~ 6
for all s' > s
=
0
I us'(Y(e» I
<e ,
•
Therefore, for the first member of the last expression of (16.33) we
have
I
I
(16.36)
J
_I
I YCe)-l(t) I ~ ~
for all s' > s
=
0
From the in probability convergence of lY(t») (s
= 1,2, ... ) to the
point l(l)' it is evident that there exists a positive integer
s~
such that
(16.37)
for all s' > s' •
=
0
It follows from (16.32), (16.33), (16.36) and (16 37) that
(16.38)
sup
I Hs'(Z(m»
z(m) € R(m)
- Hs,(Z(m» I < 3e
,
for all s ~ max(so,so)' which shows that
sup
I Hs(z(m» - Hs(Z(m» I ... 0, (s ....),
z(m) € R(m)
for, otherwise, one has always a contradiction.
(16.39)
Since the condition (16.39) is equivalent to that of (16.30), the
I
I
I
I
I
I
_I
proof of the theorem is completed.
In the case where Z(m) has the conditional pdf. ps(Z(m)I Y{j»' and
hence the pdf.
(16.40)
we define
(16.41)
hs(Z(m»
= ps(z(m) r c~»),
and denote the corresponding variable by Z{m) •
Then, the following is an immediate consequence of the above theorem.
COROLLARY 16.2.
Suppose that the condition (ii) of the above theorem
and the following condition are satisfied.
(i') For each s, ps(Z(m)
I Y( .t»)iS
over R(m) x D(l) , with D(t)=[Y'lJ
Kj's, and ps(Z(m)I Yet»~
IKj
continuous with respect to (Z(m)'Yce»
~ Yj , j=l, ••• ,t)
= 0 over R(m) x (R(t)-D(t)!
for some constants
I
I
I
I
I
I
I
_I
I
I
I
I_
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Ie
I
Then it holds that
(16.42)
as s
Z(m) -
Z(m)
(I,S)
.....
Now, in the last place, we shall consider the general case when n,
t
* O.
For this, Theorems 16.1 and 16.2 can be combined to get the
following theorem, whose proof is easy and is omitted.
THEOREM 16.3.
Let (X(n)'y(t)'Z(m)} (s = 1,2, ••• ) be a sequence of
(n + t + m)-diinensiOnal random variables, where n,t and m are fixed
independently of s.
Suppose that the following conditions are satisfied.
For the second marginals, (Y~)
(i)
= (~, •..
,:;J
(a :: 1,2, •.. )
as·
s
there can be found a sequence of real vectors,(c(J)= (cl, ••. ,cJ)J (s = 1,2, •.• ),
wi th c
a
i
>
0
for all i and a, such that the sequence
~')
=
(~/c~, ... ,~/Ct) , s
= 1,2, •••
converges in probability to the point 1(1) = (1, ••• ,1) as s....., where c(J)
is assumed to lie in the subset of R(t) on which Y(t) is defined.
:(11) For the conditional distribution of X(n) given YeJ)
= C(t)'
c(J)
being the same as in (i) above, there exist two sequences of real vectors,
- ld(n) ~ (d~, ••• ,d:)}(S • 1,2, ••• ) with d~
> 0 and te(n)= (e~, ••• ,e~)} (s = 1,2, •.• ),
and a sequence of n-dimensional random variables, [U(n) = (U~, ••• ,U:)}
(s = 1,2, ... ), which converges (I, S) to aome distribution U(n) as s ..... ,
such that
-
(16.43)
-
X(n) -
X(n)
(I,S), (s ....),
where X(n) stands for the conditional distribution of X(n) given Y(t) = c(t);
while X(n) = (~, ••• ,X
,~S.44)
:)
~ = d~
denotes the distribution given by
~ + e~ , L=l, ••• ,n.
Ct ) = Y(t)'
(iii) Conditional cdr. of Z(m) given X(n) = x(n) and Y
ps(z(m)1 x(n)'Y(l»' is continuous with respect to (x(n)'Y(£»
for any
fixed z(m) •
Then, for two sequences of random variables, (Z(m)} (s
(Z(m)}(s
= 1,2, ••• ),
(16.45)
=J
R(n) x
R(l)
Ps (z(m)
I x(n)'Y(l»
dLs(x(n)'Y(l»
with cdf. of (x(n)'Y(l»' Ls(X(n)'Y(l»), and
(16.46)
Hs(Z(m»
=J R
Ps(Z(m) I X(n),C(l»
dFs(x(n»'
(n)
s
= 1,2, ••• ,
(16.47)
with 'Fs(X(n»' the cdf. of X(n) , it holds that
Z(m)
tV
1m) (I, b) , (s ~ ..).
It should be remarked that the condition (i) of Theorem 16.2 preassumes that Ps(Z(m)I Y(l»
if the set
is defined allover the space
R(,t)' and therefore,
D(l) on which Y(l) is defined is not identical with R(l)' one has
to extend the original conditional cdf. in any way but continuously from
s
D(l) to the whole space R(,t)' A similar remark should be made for Theorem
16.3.
It is also noted that, in both of Theorems 16.2 and 16.3,
not necessarily be included in the set
which Ps(Z(m)
I Y(l»
C(l)
may
D(l) , but solely in the set on
is defined to be a cdf. of some probability dis-
tribution.
Some applications of the results obtained here will be seen in
["5, bJ.
The author is deeply grateful to Miss Dorothy Talley for her nice and
careful typewriting of the paper.
_I
= 1,2, ••• ) and
whose cdf. 's being given respectively by
Hs(z(m»
I
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
_I
I
I
I
I_
REFERENCES.
["lJ
S. Ikeda (1963), "Asymptotic equivalence of probability distributions
with applications to some problems of asymptotic independence",
I
I
I
I
Ann. of !nst. Stat. Math. Tokyo, 14, 87-116.
["2J
S. Ikeda (1965), "On certain types of asymptotic equivalence of real
probability distributions, I, Definitions and some of their
properties", UNO Inst. of Stat. M.imeo Series 455.
L3JS.
probability distributions, II, Further results on the
properties of type (5) asymptotic equivalence in the case
I
I
Ie
I
I
I
I
I
I
I
I,e
I
Ikeda (1966), "On certain types of asymptotic equivalence of real
of equal basic spaces", UNO !nst. of Stat. Mimeo Series, 465_
f:4J S.
Ikeda (1966), "On certain types of asymptotic equivalence of
real probability distributions, III, Further notions of
asymptotic equivalence in the case of equal basic spaces
and a relation between type (5) convergence and in law
convergence", UNO Inst. of Stat. M.imeo Series, 470.
["5J
S. Ikeda, J. Ogawa & M. Ogasawara (1965), "On the asymptotic
distribution of the F-statistic under the nUll- hypothesis
in a randomized PBIB design with m associate claSses under
the Neyman model", UNO !nat. of Stat. M.imeo Series, 454.
L6J S.
Ikeda & J. Ogawa (1966), "On the non-null distribution of the
F-statistic for testing a partial null-hypothesis in a
randomized PBIB design with m associate classes under the
Neyman model" , UNO !nst. of Stat. Mimeo Series, 466.
97