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ON CERTAm TYPES OF ASYMPTOTIC EQUIVALENCE
OF REAL PROBABILITY DISTRIBUTIONS
I
DEFDrITIONS AND SOME OF THEIR PROPERTIES
by
Sadao Ikeda
University of North Carolina
Institute of Statistics~o Series No. 455
Decemb.er,J..965
Con'tents
Summary and 5.ntroduction
1. Definition of four types of asymptotic equivalence
2. Pr~erties of four types of asymptotic equivalence
3. Asymptotic independence of a set of random variables
4. Some classes of transformations of random variables
pteserving the asymptotic equivalence in the sense of
tyPe (M) and (S)
5. Unsolved problems
Acknowledgements
References
This research' was supported by the U. S. Arrrry Research
Office Contract No. DA-ARO-D-31-124-G670
DEP~TMENT
OF STATISTICS
UNIVERS1"lY OF NORTH CAROLINA
Chapel Hill, N. C.
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SUMMARY AND mTRODUCTION
With the aim of application to some problems of asymptotic approximation
notion
a ! of asymptotic equivalence of probability distributions, called type I
asymptotic equivalence, has been introduced by the present author (1), basing
upon a distance defined over the class of all probability distributions which
are absolutely continuous with respect to a
space.
~-finite
measure over the basic
This type of asymptotic equivalence is regarded as an asymptotic
equivalence version of the so-called mean convergence in the usual convergence
theory, and it appears to be too severe for some situations of practical
application: For example, let us consider the following situation.
(~,X~, ••• , X~)}
s
(s
= 1,2, ••• )
Let
be a sequence of n -dimensional real random
s
variables where n is assumed to tend to infinity as s tends to infinity,
s
and suppose that, for any fixed n independently of s, the sequence
(X~, ••• ,X~)} (s
= 1,2, ••• ) converges in law to the n-dimensaionl normal
distribution with mean vector (0,0, ... ,0) and variance matrix
I, the unit
n
matrix of order n.
Then, what additional conditions will be needed in order
S2
s2
that the random variable Xl + ••• + Xn2 can be approximated, in a certain
non-trivial sense, by a chi-square distribution with n degrees of freedom,
s
asymptotically as s tends to infinity, and What conditions assure us that
S2
S2
s2
s2
a2
the random variable (Xl + ••• + Xk )!(XJ. + ••• +:JCk + ••• +X ) is approximated
s
s
ns
by a Beta-distribution with parameters (k , n -k ) asymptotically as s tends
s
to infinity?
s
Let us consider another example:
s
Let X(l) < X(2)< ••• < X(N)
be an order statistic of size N from a one-dimensional distribution of the
continuous type, and let (X(l)' ••• , X(n)) and (X(N_m+l)' ••• 'X(N)) be the
joint distributions of n lower extremes and of m upper extremes respectively.
I.
I
I
It has been shown that ifn+m
= o(.[N) ,
then these two random variables are
I
asymptotically independent in the sense of type I as N tends to infinity [1].
However, the condition on the order of magnitude of m+n would be improved, if
the requirement of type I asymptotic independence is weakened to, for instance,
a notion of asymptotic independence parallel to that of 'in law' convergence
in the usual convergence theory.
To treat these tn>es of problems we need
certain weaker types of asymptotic equivalence or of asymptotic independence
than that of type I.
The purpose of the present article is to introduce some weaker types of
asymptotic equiValence and of asymptotic independence which are useful for
practical applications, and to discuss some of their fundamental properties.
Many
unsolved questions are left open, Which will be stated in the final
section.
From the view point of practical applications, we confine o'l.U'selves,
throughout this paper, to real random variables, i. e., the 'basic spaces',
over whiCh the random variables under consideration are defined, are euclidean.
In most of the problems of asymptotic a.pproximation, we are given a sequence
of subsets of the basic spaces, whose probabilities we want to evaluate
approxiJrl8,tely by a certain other sequence of probability distributions
asymptotically, and the types of these subsets are usually the same for a
given problem of asymptotic approximation.
A sequence of classes to which
the above stated subsets belong, will be called the basic classes for a given
problem of asymptotic approximations.
Our definitions of the types of
asymptotic equivalence is essentially based upon the types of the basic
classes.
In section 1, four types of asymptotic equivalence are introduced
based upon a sort of distance between probability distributions associated
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wi"c,h each of four types of basic classes.
In section 2, some of the fundamental
properties of four types of asymptotic equivalence given in section 1, are
stated where the strongest two notions are shown to be mutually equivalent,
and to be equivalent to that of the type I asymptotic equivalence given in [1]
in some special case.
Section 3 is devoted to introduce two types of
asymptotic independence of a set of random variables, which is done by specializing the weaker two notions of asymptotic equivalence given in section 1.
Some of the fundamental properties of these notions of asymptotic independence
are also discussed.
In section 4, some classes of transformations of random variables which
preserve the properties of asymptotic equivalence in the twoleaker types, are
discussed.
In the last section, we list some open questions which are
important for practical applications.
1.
Definition of four types of asymptotic equivalence
Let !R(n) be the n-dimensional euclidean space and <B(n) be the :Borel
field of subsets of !R(n).
Denote the family of all probability distributions
defined over the measurable space (!R(n)J (B(n» "by 3(!R(n),<B(n»' a member of
which is designated by a random variable X(n)
or equivalently by a probability
measure pX(n) , according to which X(n) is distributed.
Let C(n) be any subclass of <B(n)' and let us define, for X(n) and Y(n)
belonging to 3(m(n)'~(n»' a quantity associated with the sUbclass C(n)
(1.1)
3
I
Clearly, this is a metric over ~(n)' CB(n»
if we identify those random
variables which have the same probability mesure over C( )' i.e.,
X
y
p (n)(E(n»
n
• p (n)(I(n»
Let us define
for all B(n) belonging to C(n).
tbr.~
of eB(n) CGnsisting of
&1.1
subclass of (B(n) as follows: Men) is the subclass
subsets of M(n) which have the following form
i = 1,2, ... ,n)
,
Sen) is the subclass OfCB(n) consisting of all subsets of R(n) which have the
following form
i = 1,2, ... ,n) ,
and finally, Q(n) is the finitelY additive field generated by Sen) or equivalently by
.H(n).
Then, it is clear that the implication relatiClls
hold and Q(n) is constructed from Sen) in the following manner:
k
i
i
Q(n) • ( U I(n) I E(n) E Sen)' i
i=l
.
= 1, ... ,k,
k is any finite positive integer)
Now, let us consider two sequences of random variables £X(n»)(s = 1,2, ••• )
. s
s
8
s
and Jf(n ) ) (8=1,2, ... ,) with X(n ) and Yens) belonging to 3(M(n ),CB(n » for
s
s
s
s
each I, where n. mayor may not depend on s. The case where n. is fixed
independently of s will be called the case of equal basic spaces.
Another
-
important case i.: that where ns tends to infinity as s tends to infinity.
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!
~­
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types of sequence of basic classes: (~n ) }(s=1,2, ... ), £S(n )}(S=1,2, ... ),
s
s
fa(n ) }(s=1,2, ... ) and tB(n ») (s=1,2, ... ). Let £C(n ») (S=1,2, ... ) designate
s
s
s
any of these four sequences:
DEFINITION 1.1·
Two sequences of random variables (X(n ) }(s=1,2, ... )
s
s
and (Y(n ) }(S=1,2... ) are said to be asymptotically equivalent in the sense
s
. of type (C) as s.-. eo and are denoted by
J
1
1
1
cl
Corresponding to these sequences of random variables, we give four
e
]
1
]
(C)
if
We shall call this type of asymptotic equivalence the type (C) asymptotic
s
s ) are asymptotically equivalent ,C
l )
equivalence, and say that X(n)
and Yen
,
s
s
briefly.
Identifying (C(n ») (S-1,2\O •• ) to each one of the four sequences
s
(~ns).) (s-1,2, ... ), (S(n »)(S=1,2... ), h(n »)(S=1,2... ) and (lB(n »)
s
s
s
(s=1,2~ .. ), we get four types of asymptotic equivalence, type (H), (8), (a)
and (lB).
For these four types of asymptotic equiValence, it is clear from (1.2)
]
that the implication relations
J
(1.4)
J
hold over 3, i. e., for any two sequences (X(n ») (s=1,2... ) and
(eB) ==> (a) ==> (8) ==> (I()
8
(Y(n »)(8=1,2••• ) with X(n ) and Y(n ) belonging to 3(m(n ), eB(n » for each s •
J
8
S
S
S
]-
1
5
S
-I
It should be noted that the type (m) asymptotic equivalence is an
equivalent notion to that of the type I asymptotic equivalence given in [1]
. under the situation in which x(n ) and Y(n ) under consideration belong to
s
s
the f'am1ly P{M{n
men ), Y(n » , the family of all »robabUity distributioo.s
s
s
s
wtILch are absolutely continuous with respect to a preassigned a..finite
»,
measure Y(n ) over (M{n ), men
s
s
s
».
In fact, under such a situation we
have
for each s, where f (n ) and g(n ) stand for the generalized probability
s .
s
density function of x(n ) and Y(n ) with respect to Y(n ) respectively, and
s
s
s
our previous notion of the t=roe I asymptotic equivalence was based upon the
distance Which is the right-hand member of the above equality.
2.
Properties of four mes of asymptotic equivalence.
In the present section, we shall discuss some of the fundamental
properties of four t=roes of asymptotic equivalence given in the preceding
section.
First, it is clear that
IEMMA. 2.1.
In general, the t=roe (C) asymptotic equivalenc~ is
transitive in the sense ,that, if X(n ) .... Y(ns){C(n » and Y(n ) .... Z(ns)(C{n »'
s
s
s
s
then x(ns)~Z(ns)(C(ns».
Thus, each one of four types of asymptotic
equivalence given in the preceding section
lEMMA. 2.~.,
Let (C(ns)~ (s=1,2,
haS
this property.
) be any one of .t~~ four secpences
(H(n ) }(S=1,2, .. ,), {S(n )} (s=1,2,
), (a(ns)} (s=1,2, ... ) and
s
s
iB(n ) }(S=1,2, ••• ). Then, x(n ) .... Y(n ) (C(n ) implies that
s .
s
s
s
,
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where X(n) and Y(n ) are marginals of X(n ) and
s
s
s
xs( )
ns
for which the choice of
=
(X~ , ... , x~ ) and YS(n ) = (~ , ... , Y~ ),
m
1
{i , ... , i }
m
l
s
1
m
s
out of {1,2, ... , n } mayor may not
s
depend on s •
The following result would be helpful for practical applications of
our notions of asymptotic equivalence.
LEMMA 2.3.
In order that X(n )""Y{n )(C(n
s
s
s
»
as s ~IO, it is
necessary and sufficient that
I
e
'J
1
1
XS
yS
s
) - p (nS)(ES(n »
p (ns)(E
(n s )
s
J.
0 , (8
~IO)
{E(n)} (S=1,2... ) with E(n )€ C(n ) for each s. Thus,
S
s
s
this is true for each of the four types of asymptotic equivalence 'given in
the preceding section.
The proof of this lemma is quite similar. to that of Lemma. 1.3.2 of
[1] and is omitted.
In the next place, we shall discuss the implication relations of our
four types of asymptotic equivalence.
First, we can show the following
THEOREM 2.1.
J
1
I~
for every sequence
]
It :OOIds that
(n) <'-=> (m)
(2.2)
over :J •
PROOF.
This is proved by making use of the so-called extension
theorem [2].
J
1
s
s
7
I
To .prove
.-I
the theorem, it is sufficient to show that (a) =as> «(B).
(£~ }(s-1"2,, ... )
Let
be a sequence of positive numbers such that
s
fhen" for each s, there exists a member of (B(n )' E(n )
s
s
S ... 0 as s ... eo.
say" such that
E
o< ~
-
(n )'
s
(~(n
S
),i:(n »s
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~
i:
Ip (ns)(E s
) - p (nS)(E s
)1 < E
(ns )
(ns )
s
the extension theorem,. ~ne can find two coverings of E ),_
ens
ren )1 = t!'(n )11 }(i=1,2" ••• ) and r(n)2 = (Fen )2j) (j=1,,2, ••• ) with
s S
Ss
s
s·
F(n )11 and F (n )2j belonging to a(n) for all i and j, such that
!y
s
B
S
.
and
I
,
_I
where, we can assume" 'Without loss of generality,
-S
r (n ) =
s
I'IIIs
1,&
(n )li
B
S
n , (nS)2j
and rearranging them, let r(n )
s
B
S
that F(n)k
s .
s
F(n )
s
eo
II
n '(n
S
)k' =
S 10
lomB
1& (n
= (len
S
S
B
S
)li E.T (n )1" ' (n )2j E r (n )2' for all i and. j),
B
S
)k) (k=l"~,, ... ).
S
Then, it is clear
s.
B..s
¢ (k F k') and l'(n) is a covering of E( ). Since
10
S
ns
S
•
E F(n )k" I:, (n )li and E F (n)2j are members of (B( ) and
k=l •
i=l s
j=l.
n.
s
, (n ) S
s
(10
S
,(10
S
Ei=l' (ns )li'l n Ej=l' (ns )2j
we have
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)
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!
~
r;
(2.5)
yS
~
P (ns)(F s
) < ZCO P (ns) (F s
j=I
(n )2j
(n)
s
S -
and
)
,
end,o:f course
xS
S
~
~
) < p (nS)(F s
)
(n s )
(n ) s
X
) < P (ns)(F s
P (ns)(E s
)
(ns ) (ns )
J
]
]
)
) < 10 x(1S) ( S
P (ns)(F s
F (n )li
(n) - Zi=lP
S
S
end
p (nS)(Es
Hence, it :follows :from
(2.4), (2.5)
•
and
(2.6)
that
1
]e
-}
]
1
'1
J
s
COs
Jl S
Now, since F (n )::: Z F (n )k' and ~=J! (n)k is a member o:f
S
k=l S
S
for 8.1'lY fixed N, we can find a positive integer N' = N(s) such that
~
P (n s ) (Fen
s
(2.8)
» -
rl
£S < P (ns )(
N
Z Fen )k)
k=l S
s:1multaneously.
Hence, it follows :from (2.7) that
:from which we get
]
1
~
(ns ) (Fen
»
S
~ .
~(N
~(
P (ns)(F s
) < P nS)(F s
)
) _ € < P n s )( ZF s
(n s )
k=l (ns)k (ns )
S
J
J.
=s p
9
end
a (n
S
)
(2.9)
(X(n )'Yen »-+0 (s -+10) implies that
)
s
s
(s -+ 10), which c&itpletes the proof of the theorem.
Therefore,
8
a(n
8(9
(n )
s
(X(n )'Y(n »-+0
s
s
Equivalence relation between the type «(I) asymptotic equivalence ani
the type (8) asymptotic equivalence is not valid for many practical cases:
For exanple, let (X(n )} (s=1,2, ... ) be a sequence of random variables of the
s s
discrete type, WhUe
(Y(n ) }(s=1,2... ) be those of the continuous type.
s
Then, for these two sequences, the type (<8) asymptotic equivalence never
holds.
It would be useful to give conditions, under which the two types
of asymptotic equivalence, type (<8) and type (8), are mutually equivalent.
we
do not enter into further discussions of the properties of the
type «(I) asymptotic equivalence, because we have done this already in the
previous paper [1] for some practically important cases.
Now, we shall consider the implication relation between tw notions
of asymptotic equivalence, the type (J() and the type (8).
First
'We
show the
following
THEOREM 2.2
If
n
s
is bounded above uniformly far all s, then it
holds that
V4Q <-=> (8)
(2.10)
over :J.
PROOF.
It suffices t.o show that (,4(,) -> (8).
Let, for each s, E (n )
s
Then, there can be found a positive integer
be any fixed member of 8(n).
n
s
Ns(S 2 s) and a sequence
£F(n )i} (i=1,2, ... ,Ns ) such that
s
10
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and
where
s
ci
=+ 1
or - 1, i
=1,2, ••• ,Ns
•
Hence we have
(2.11)
from which the theorem follows.
:By this theorem it is known that the equivalence relation (2.10) holds
true for the cases of equal basic spaces, and hence, for convergence cases.
The following example shows that the equivalence relation (2.10) can
not necessarily hold in general case.
EXAMPlE 2.1.
Let
JS., ...,
b"~
In
a random sMlPle of size n drawn
from the uniform distribution on [0,1) , while ~, ... , ~ be that from a
distribution with prObability density function given by
{
lin,
if -1 ~ z
<0 ,
l-l/n, if 0 ~ z < 1,
0,
(n=1,2, ••••••• )
otherwise.
Then, for the set E(n) in B(n)' defined by
E(n)
we obtain, for
X(n) =
x( )
n (E(n»
Ip
= [0,1)
(JS., ... ,
ti(n)·
.. P
(E(n»
x [0,1) x .... x [0,1) ,
Xn) and Y(n)
I=
= (~ , ...,
1
1 .. (1 .. ii)
11
n
y~) ,
1
... 1 .. e- , (n "'10)
I
'Which means that x(n)
Y(n) (S(n»' (n -+ 10) can not hold true.
N
On the other hand, we can easily see that
8~n) (X(n)'r(n» S
lin,
n=1,2, ••• ,
'Which implies that X(n) N Y(n) (~n»' (n -+10 ) •
3.
AsYD!Ptotic independence of a set of random variables.
In the present section, we introduce two types of asymptotic independence
of a set of random variables, by specializing.phe notions of asymptotic
equivalence of the type
4AO
and the type (8).
The method is quite similar
to that of the previoUs paper [1].
s ) = (a
a
a )
Under the same situation as in section 1, let X(n
Xi,X2'."'X
ns
s
be a member of 3(m(n )' ,(B(n », s=1,2,.... Corresponding to a decomposition
a
a
Ofm(n ) in the product form:
s
m(na) = m(lD.;t) x m(m ) x ••• x m(IIlt)
s
s
let us write
,
X(n) in the form
B
(3.1)
x(n )
s
= (X(lD.;t)'X(~)'
... , X
»
CIIlt
s
'Where X(m ) belongs to :J(m(m )' (B(m » for each j and s, arxl lD.;t,~, ... ,IIlts
j
j
j
and k mayor may not depend on s under the restriction that mj
a .
~
+
~
+ ••• +
~
B
= na •
For each
B,
we consider the set of marginal variables
of x(n ):
s
... ,
12
> 1 and
-
•
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..
•
w
Let us consider, for each s, a member of ~(m(n ),<B(n »
s
s
... ,
Y(~ » ,
s
whose margina.ls
yS
k
P (nS)(E
(n s )
) =
S
n
i=l
yS
p (mi)(E
for any E(n ) = E(~) x E(~) x ••• x E(~)
s
s
(mi )
)
with E(mi)€<B(m )' i=l, ••• ,k s •
i
It is known that for any given X(n) there exists a Y(n)' having the
s
s
property stated above, such that
(:3.6)
for every E(.
~
) belonging to <B(
~
), i=1,2, ••• ,k
s
Under this situation, we give the following
DEFINITION 3.1.
•
A set of random variable (3.2) is said to be
as:vn!!?totica.lly independent in the sense of type ~}. as s -+ to, if two
sequences
(X(n» (S=1,2, ... ) and (Y(n» (s=1,2, ... ) are asymptotically
s
S
equivalent in the sense of type ~) as s -+ to •
Likewise
DEFINITION 3.2.
A set of random variable (3.2) is said to be
asyn;ptotically independent in the sense of twe (8) as s-+ to, if
(x(n » (S=1,2, ... ) and (Y(nsl) (s=1,2, ... ) are asymptotically equivalent in
s
the sense of type (8) a.s s -+ to •
I
In these two cases, we shall call the set (3,2) an as:y1!ftoticaJ.ly
independent ,,) set of random variables and. an asymptotically independent
,(§) set of random variables, respectively.
(S) implies G,At) in generaJ., it is straightforward that
;.1. If. the set of random variables (3.2) is asymptotically inthe sense of type (8) as s-..o, then it is asymptotically independent
.-.
of type ~ as s4 co.
We list some fundamental properties of the notions of asymptotic
Since
LEMMA
dependent in
in the sense
independence given above in the following leIlIll'J8S, Whose proofs are easy and
will be omitted.
IE:MM.ll3.2.
'Where the choice of
depend on s.
(X(m)' ••• , X(m , , ) } be,,~y subset of (3.2),
ol
'''il
.L
{i~, ... , i.t } out of s {l, s, ... ,Its} mayor may not
Let
s
Then, the asymptotic independence Ve) ( (S»
of
X(~ ) } implies the asymptotic independence ~)
s s
X(m )} •
i.ts
LE~ 2.3.
(X(n) =
If
tx{m )' ...,
«Sn of
X(~ )} (S=1,2, ••• ) and
s
s
s
S
1
ks
(Z(n ) = (Z(w,)' "" Z(~»} (s=1,2... ) are asymp'tot,i,C,ally equivalent in
s
the sense of type ~) «S) as s .... eo , and if (Xeml)' ••• , Xc
)} is
f
asymptotically independent in the sense of type (M)
then
«S»
llJts
as s .... eo
,
(Z(ml) , ••• , Z(~s)} (S=1,2••• ) are asymptotically independent in the
sense of'type
W> «S»
LEMMA 3.4.
as
s .... eo.
If n
s is bounded above uniformly for all s, then both
types of asymptotic independence, type ~) and t~e (S), of the set of
random variables (2.3) are mutuaJ.ly equivalent.
The following result is also immediate.
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lEMMA 3.5.
Let
f~(X) 's be measurable transformations fran !R(l) into
!R(l)' and
u(n )
s
= (u1s
, •••-,
US )
n
s
U~~ = f~(~i)
~
'
i=l, ••• ,ns
and
S
vS(n ) = (~l ' ' ' ' ' V ),
s
ns
(3.3)
Vis = f~(r),
~
~
i=l, ••• ,n ,
s
Suppose that x(n) NY(n) ("n » (s ~IO) implies that
s
s
s
v(n ) (~n » (s ~ 10), that is, the function
s
s
for every s.
u(n )
s
N
preserves type (.&() asymptotic equivalence.
of (~, ""
Then, the asymptotic independence
X~s) in the sense of type (J() implies that of
{U~, ••• , ~s}
in the same sense.
The same is true for type (5) asymptotic independence.
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•
4.
Some classes of transformations of random variables preserving the
asymptotic equivalence in the sense of type
decide the classes of transformations of random variables preserving the
type (.A() and type (5) asymptotic equivalence.
same
Given two sequences of random
(Yens)} (s=1,2... ) with X(n )
s
belonging to :J(m(n ), CB(n » for each s, let
s
s
•
-.
In this section, we discuss
simple classes of such transformations.
variables, oc(n )} (s=1,2... ) and
s
-e
and (5)
From the viewpoint of practical application, it is inportant to
The problem is formulated as follows:
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and Y(n )
s
I
(4.1)
where f(ns,m ) (Z(n >" is a measurable transformation fran mens) into !R(m ) ,
s
s
s
(n > m ), for each s. Then, what conditions should be imposed on the
s- s
,,,
8
function fS(n m) and 'on the sequences (X (n » (s=1,2... ) and £rl(n »
.. ,
8' 8
S .
s
(8=1,2... ), in order that x(n ).... Y(n )(~n» (s -.eo) implies
•.8
• .8
S
8 S
8..s
V(m )"" Vema) (1:n » (S-'IO), or that X{na)N I(n )(S(ns»(S-'eo) implies
s
s
s
u(m )"" v(m )(1n (s-.eo) and/or u(m ) "" V(m )(S{n »(S-'IO)1
S
s
s
s
s
s
Y
This problem would be very difficult to solve in the general case.
In
the present section, we consider some special cases and discuss this problem.
lSefore entering into our discussion, we shall state tll:! following well-known
result.
equiva.lence of oc(n) }(S=1,2... ) and Yen) (i.e., the type ~) convergence in
this ca8e)
implies that of u(m) = f(n,m)(X Cn » and V(m) = f(n,m)(Y{n»
the S&1IE. sense, if f( n,m) is independent of s and is continuous.
Now, in the first place, we consider the case when
,
.
where z(n) = (zl' ... , zn ) and c~ts and d~'S are constants such that
s
's
c~ > 0 for all i and s.
Then, we can show the following
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4.1.
Suppose n = n and m = m (m < n) far all s, and
s
~
Y(n)'s are aJ.l identical t~ ~~ fixed Y(n).. Then the type (M) asymptotic
LEMM.'
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implies that U(n ).... v(n )(S(n »(s~IO). The converses are also true.
s
s
s
PRCXlF.
This theorem follows easily from the facts that
and
for each s.
In fact, these imply that
s
·81t11
(x ( );i(! » = 8 ~
(Us
•.8
)
~,n)
ns
ns
-,n) (n )'V(n )
s
s
s
s
and
respectively.
As a special case, let us consider a case of equal basic spaces for
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which
n
c~
(i=l, ... , n; s=l, 2... ) can be replaced by the conditions c ~
1
>0
s
= n (s=1,2, •••,),., Then, it is easy to see that the conditions
1
s=1,2, ... ).
f
0 (i=1,2, ... , n;
This gives a theoretical foundation to the so-called "asymptotic
distribution": . For example, _. let
Us
and Y be random variables distributed
according to a binomial distribution :B(s,p) and the standard normal distribution N(O,l).
Then, under the limiting s~1O , p~ 1\(~ 0), and Sp~1O ,
X = (U -sp)/~SP(l-P)
s
s
tends to Y in the sense of type (S),and therefore, by
the above theorem (for n
. ,
s
.
= n), (Us) (s=1,2... ) and (VS =.Jsp(l-p) Y + sp)
(s=1,2... ) are asymptotically equivalent in the sense of type (S) under the
same limiting.
~
For sufficiently large s,
of Us •
17
V
s
is called an asymptotic distribu-
I
Next, let us consider the case of equal basic spaces, and lBt
... , 1) ..
~
for all ,zen) = (zl'"'' Zn) belonging to m(n)'
Then, we can prove the
following.
mOREM 4.2.
Suppose that Y(n)'S are absolutely continuous ,with
respect to the Lebesgue measure J.I.(n) over
~
given
€
> 0" there
men)
uniformly for all s, i,e., far
ex:tsts~pOSitive constant
that, i f tlcn) (B(n» < 8 then pen) (N(nY< E
,
8
~
for all s.
depending on s such
Then under the
situation stated above" if x(n) -Y(n) (Hcn»(s....)" then u(n) -v(n) (1n»(S~to) •
~.
* be the class of all subsets of men) of the forms
.Let Sen)
i=l, ••• ,n)
and
Then, under the B.$sumption of this theorem, it is easy to see that the t'WO
conditions 8 ) (x(n)' Y(n»
8fn
mutual.ly equivalent.
~O (s~ to) and 8S(n) (x(n)'Y(n~"'O (s~to) are
Now, since
=
{ Zi1zi < 0 }
{
zil~ < zi SO}
i
18
, if &i = 0
, if a i < 0
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for each i, i=l, ••• ,n, the set
is
e~ressed
in the form
N *
= p=l
1: E )
(n j
Sen) (j=l, ••• ,N) and N = N{al, ••• ,an ) is a certain positive
integer not greater than 2n uniformly for all (al, ... ,an ). Hence, it follows
where Een)j
€
that
whence the theorem follows.
Let us consider another situation where
(4.4)
are such that each f~(Zi) is a monotone increasing function of zi' i=1,2, ••• ,n ;
s
s=1,2... Then, clearly
-1
fS(n
n )(Ml »
s' s ., ns
s
-1
.M{n ) and fen n )(S(n »)S S(n )' s=1;2••• ,
s
s' s
s
s
from "Which we obtain the following
THEOREM 4.3.
implies that U(n)'" v(n )(.Hen » (s ~eo).
s
s
s
asynq>totic equivalence.
In the last place, let us take a transformation from !It(2) to !It(l)
defined by
I.
I
..-.
-
Xens )'" Y(n ) (.~n y<s-+ eo)
S
S
The sa.:roo is true for type (S)
Under the above situation,
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c(~
0) being any fixed constant" and suppose that two sequences of random
variables (X(2) = (~"X:)} (S=1,,2... ) and
(1(2) = (~,,~)} (s=1,2... )
satisfy the following conditions: .
(i)
One of
: the marginals of Y(2)' ~ s8\Y tends to a certain
distribution Z of the continuous type in the sense of type ~) as s -+ eo • .
(ii) 1(2)'s are absolutely continuous with respect to the two-dimensional
LebeS~e measure over (!R(2)" CB(2» uniformly for all s.
Then" ·we can show the following
BOREM 4.4.
then
uS.... yB
Under the above situation" if X(2).... Y(2)~2»(s -+110)"
(.Hel»(S -+eo)" where
(4.6)
PROOF.
:By the condition (i)" for any given e;
> 0 , there exist a set
E~ e; S(l) and a positive integer Se; such that
~
11 - P l(E~) 1 < e; and
(4.7)
for all s
> s e; • Put
-
Fig. 1
20
~
11 - P
(E~) 1 < e;
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Dividing E~ into N subintervals, and giving a straight line
zl + cZ2 = a,consider the N members of 8(2)' E{2)(a),
in Figure 1. Then, we can choose N such that
E(2)(a)""'~2~a) as
independently of the value of a, and hence, there exists N
= N(e)
such
that
and
< a)
uniformly fat' a and for sufficiently large s, where E(a) = (zlz
I pif (E(a»
(4.8)
-
p~ (E(a» I :s I
XS
p (2)
N
yS
(i~l E(2) (a) )-p
(2)
€
fi(l)'
N
(i~lE(2) (a» I + 46
uniformly for a, and for sufficiently large s, from Which we have
for sufficiently large values of s.
COROLLARY 4.1.
This results the theorem.
situation of the above theorem, suppose
Under tl'J3
x~ and Y~ are positive random variables 1=1,2., and let
if =
(4.10)
Then,
~
~
S
and V
=:i .
y~
x(2).... Y(2) ()1(2»(s ~eo) implies that
PROOF.
Us.... ~(12»(s ~eo) •
Applying Theorems 4.3 and 4.4, we have
21
(4.11).
Hence" using Theorem 4.3 again"
'We
have
wbich proves the corollary.
We conclude this section by the following eX8Jl';?le.
EXAMPLE 4.1.
(i)
Suppose that
£X(2) = (~,,~) }(S=1,2... ) and tt(2) = (~,~)} (s=1,2
asymptotically equiwJ.ent in the sense of type Vi) as s
(ii)
) are
10 ,
x~ and ~ are positive random variables, i=1,2; s=1,2, ••• ,
(iii) . ~ tends to a certain distribution Z of the continuous type in
the sense of type (ft.), i.e., in the sense of 'in law' convergence,
as s .... 10" and
(iv)
Y(2) 's and y~~ts are absolutely continuous with respect to the
Lebesgue measure over (!R(2),,<B(2»
uniformly for all s.
Put
(4.~)
, s=1"2" ••• ,,
then it holds that
(4.14)
In fact, from Corollary 4.1 and Theorem 4.1 it follows that
.1.. = 1 + ~
t!
.~
N
!.. = 1
vB
+
Using ~rem 4.3" we have (4.14).
22
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(.M( », (... e) •
~
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5. Unsolved Problems
To establish the theory of aBYIIi>totic equivalence in the sense of type
(M) and (8) which is enough for practical applications, many questions are
left open.
We list some of them below.
(a) . First of all, it is desirable to give certain criteria for type
ex) and type (8) asymptotic equivalence in both of the
cases of equal and
unequal basic spaces.
(b)
l'obre general results are desirable for the problem stated in the
beginning of Section 4, i.e., the problem of determining the class of m.easurable transformations 'Which preserve type (X) and type (8) aSYIli>totic equivalance property.
In practical applications, we sometimes meet the following types of
(X(n )} (S=1,2... ) and CX(n » (S=1,2... ), the two
s
s
sequences of k s marginals of them, (X{k ) = (X~ , ••• , X~ » (s=1,2, ••• )
r.:S
s
s
s
1.
K S
and tY(k ) = (Yil'''.'Y~ }(S=1,2, .._.) are aBymptot:1.cal~ equivalent in the
s
sense of type (M) or (8) s as s -+ 10, where k < n and the choice of
s - s
(il , ... , ~s) out of (1,2, ... ,ns ) and k s mayor may not depend on s. Then,
problem: (c) For
what conditions will be needed for the validity of u(m) ~ V(m )(}'t(m \)(s -+10);
far u(m)
s
s
and
v(m) given by (4.1), or (d) for cx(nji
s
(S=1,2~•• ) ~d
(Yen » (s=l,2, ... ), for any given finite integer k the t't'TO sequences of any
s.
...::s
s
s
s
_.8
k ma.rgmals of them, tX'(k) = X. , ... ,X. } (s=1,2, ... ) and ~(k) = (Y. , ... ,I. )}
:1.1
:1.k
:1.1
~
(s=1,2, ...), are aSYIli>toticaJ..ly equivalent in the sense of type (.M.) a.s s -+00 ,
where (i , ... , ~) is dependent or not dependent on s.
1
Then, what conditions
will be needed for the validity of u(m) "" v(m )(Mcm
s
and v(m ) given by (4.1).
s
s
s
» (s
-+10), for U(m )
s
I
(e)
Suppose we are given a sequence of random variables
«x(n )-' t(m »} (S=l,2, ... ) such that the first marginal x(n ) is of the
s
s
s
continuous type, whose conditional probability density function given
t(m)
= Y(ms )
»,
being ps(X(n )IY(m
and for the second marginal Y(m)
s..
s
s
.
s
there can be found another probability distribution Z(ms) which is .
s
asymptotically equivalent to Y(m ) in the sense of type (8) as s -'10 •
s
!L'hen, What conditions are sufficient in order that two probability distributions x(n ) and f(ns) are asymptotiCally equivalent in the sense of
s
type (M) or (8) as S-'IO, where X(n) is a random variable 'Whose probability
s
s
density function being given by taking the expectation of ps(X(n ) Iz(~s»
s
s
with respect to Z(n ). Here, n and m mayor may not depend on s •
s
s
s
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AC:KNOWLEDGEMERrS
The author is deeply gratefUl to Professor N. L. Johnson for his
vaJ:uable comments by which this paper was revised.
Mt's. D. Lykken for her nice typewriting.
Thanks are also due to
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REFERENCES
[1]
S. Dteda (1963), "Asymptotic equivalence of probability distributions
with applications to same problems of
Ann. of Inst. Stat. Math., Vol.
[2J M.
w:ve (1955);
asy~totic
J:.i, 87-1l6.
ProLability Theory, Van Nost.
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25
independence",