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ON CERTAIN TYPES OF ASYMPTOTIC EQUIVALENCE
OF REAL PROBABILITY DISTRIBUTIONS
II
FURTHER RESULTS ON THE PROPERTIES OF TYPE (I!)
ASYMPTOTIC EQUIVALENCE IN THE CASE OF
EQUAL BASIC SPACES
BY
Sadao Ikeda
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No.
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March 1966
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465
Contents
Summary
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1.
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2.
Some of the fundamental properties of type(.$) as;ymptotic
e quivalence.
On a class of measurable transformations preserving type(S)
as;ymptotic equivalence.
3. As;ymptotic equivalence(S) of the marginal random variables.
This research was supported by the U. S. A:rrrry
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a-
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Research Contract No. DA. ARO-D-3l-124-G670.
Department of Statistics
University of North Carolina
Chapel Hill
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Summary
The author introduced in [lJ some types of asymptotic equivalence
of real probability distributions and discussed some of their properties.
the present article some additional results are stated on the properties of
one of those types of asymptotic equivalence, -type(.S), in the case of equal
basic spaces.
In this case, type (S) and type (J() asymptotic equivalence are
the mutually equivalent notions, and hence the results obtained here are
applicable equally to type (M) asymptotic equivalence.
In the first section, some fundamental properties of type
(S)
asymptotic equivalence are discussed in connection mainly with the usual in
probability convergence.
Section 2 is devoted to show that a finite linear
sum of the marginal random variables preserves the type (~) asymptotic equivalence of the original sequence of random variables.
In section 3, some results are shown on type
(S)
asymptotic
equivalence of marginal random variables when the other marginals are replaced
in a sense by another random variables which are asymptotically equivalent (S)
to these marginals, or when some of them converge in probability in a certain
manner.
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In
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1. Some of the fundamental properties of type(S) asymptotic equivalence •
Throughout this paper, the dimensions of the basic spaces are
fixed independently of the limiting process under consideration.
As for the terminologies and definitions of type (j) or type (M)
asymptotic equivalence, reference should be made to [lJ.
In the first place, we shall define two kinds of property of a
sequence of real random variables, one is a sort of uniform. absolute continuity
of a sequence, and the other is a sort of boundedness in probability.
Let {x{n) = (~, ••• ,
x:)
l(s
= 1,2,
••• ) be a sequence of
n-dimensional random variables defined over the euclidean space (!R(n)' eB(n)'
~(n))' where eB(n) is the Borel field and ~(n) stand for the ordinary Lebesgue
measure over (m(n)' eB(n)).
For this sequence we shall give the following definitions.
DEFINITION Ll Let N be any given positive integer.
Then, a
sequence of random variables, {x{n)}(s= 1,2, ••• ), is said to have ~
property eN(s ) , if, for any given
positive integer s such that
o
sup
(1.1)
E(n)
€
€
> 0, there exist a constant 6 > o and a
x:;
P (n) (E(n)) < €
.8(n)(N,6)
for all s not less than so' where ~(n)(N,6) is a class of subsets of R(n)
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defined by
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i=l, ••• , N} •
-,
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DEFINITION 1.2 (a)
A sequence of random variables,
{X(n)} (s = 1,2, ••• ), is said to have the propertyB(S), if, for any given
€
>
0, there exist a positive integer So and a member, E(n)' of S(n)' whose
closure being compact, such that
for all s not less than so' where E(n) does not depend on s.
(b)
If, in the above definition (a), the subset E(n) is allowable
to depend on s under the limiting, i.e., if, for any given
such that
(1.4)
for all s not less than so' then we say that the sequence (x(n) }(S=1,2, ... , )
has the property B*(P).
For these definitions, it is easy to see the following results.
LEMMA 1.1 (a)
If N'
< N, and if {x(~) }(s=1,2, ... ) has the
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equivalent ($) as soo)
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> 0, there exist
a psoitive integer So and a member, E(n)' of p(n)' whose closure being compact,
property CN(~)' then it has the property CN' (~).
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€
(b)
If {x(n) }(s=1,2, ••• ) and {Yen) }(S=1,2, ... ) are asymptotically
cx),
and if one of them has the property C ($), then the
N
other has the same property, for any given N.
(c)
If {x(n)}(S=1,2, ••• ) converges in law to some probability
distribution of the continuous type (this is equivalent to the type
vergence), then {X(n)}(S=1,2, ••• ) has the
3
(S)
con-
pr~perty CN($) for any fixed N.
(d)
Let (X(n)}(S=1,2, ••• ) and (Y(n)}(S=1,2, ••• ) be two
sequences of random variables such that
..s
..s)
s
s s
s
Y(n) = ( 1 1 , ••• , In ' Yi = ci Xi + di , i=l, ••• , n,
where
c~ ~ c~, for some positive constant c~, i = 1, ••• , n.
Then, the
property CN(~) of (x(n)}(S=1,2, ••• ) implies that of (Y(n»)(S=1,2, ••• , )for
every fixed N.
(e)
If the sequence (x(n)}(S=1,2, ••• ) is such that the
variables
s=1,2, •••
converges in law to some probability distribution of the continuous type, then
the sequence
has the property
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CN(S) for any given N.
LEMMA 1.2. (a) I f (x(n)}(S=1,2, ••• ) and (Y(n) }(s=1,2, ••• )
are asymptotically equivalent ((J) as s-too, and if one of the sequences has
the
propertyB(~),
then the other has the same property.
The same holds for
the property B*($) too.
(b)
If
(x{n)} (s=1,2, ••• ) converges in law to some probability
distribution as s"* 00, then the sequence has the property B(,9).
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(c)
Suppsoe that the sequence (x{n»)(S=1,2, ••• ) has the
property B(~), and let {(c~, ••• , c:)} and {(d~, ••• , d:)}(S=1,2, ••• ) be two
sequences of real vectors.
Let us consider the sequence (Y(n)}(S=1,2, ••• )
defined by
dS i =,
1 ••• , n
Ys(n) = (ysl' ••• , yS)
n' ySi = c is.~
~ +. i'
(1 •5)
Then, if
n
s 2
n ( s 2
(c.) and E d )
i=l J.
i=l i
~
are both bounded uniformly for all s, the
sequence (y(n) }(s=1,2, •••. ) has the property B(;;).
has the property B*(P).
(d)
s
/
If (X(n)}(S=1,2, ••• ) has the property B*(~), then, there
can be found .two sequences ((c~, ... , e:) }(s=1,2, ... ) and (d~, ., •• ,d:)}
(s=1,2, ••• ) of real vectors such that the sequence (Y(n)}(S=1,2, ••• )
defined by (1.5) has the prop~rty B(~).
Now, we shall
~how
the following
THEOREM 1.1 Let ((X~, X~)}(S=1,2, ••• ) be a sequence of twodimensional random variables such that
(i)
(11)
the first margina1s, (X~}(S=1,2, ••• ), has the property
the second margina1s, (X~}(S=1,2, ••• ), converges in
probability to some constant A.
Then, it holds that
(1.6)
,_
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In general (y(n) }(s=1,2, ••• ) .
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PROOF:
p(~
+
~
< a) =
I
Since
P(~
<
I
a-X~, X~-AI
>6) +
p(X~
<
I
a-~, ~_AI
..
..........'
<
6)
"
and
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for any given 6> 0, we have
Since the sequence (x~)(s = 1,2, ••• ) has the property C (~), there eXist,
1
for any given e > 0, a constant 50 > 0 and an integer So > 0, such that
sup
p~
(1.8)
Hence, if we choose 6in (1. 7) such that 2 6 < 60 , then
sup p(a-A-5~ ~ < a- A+5)
-
- ex> <
<
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(E) < e
E e ~(1) (1, 50)
for all s ::: so.
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e.
a<oo
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1"
s
On the other hand, since X converges in probability to A as
2
s ~ 00, there eXists, for such a choice of 6 as above, a positive integer s'
o
such that
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pel X~-A.I
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> 6) <
€
for all s > s'o •
From (1.7), (1.8) and (1.9) it follows that
sup
(1.10)
P(X~ + X~ < a) - p(X~
+ A. < a)1 < 2
€,
-co < a < co
for all s > max (s o , s'),
which means that both of the sequences in (1.6) are
0
asymptotically equivalent (J() as s-;.co, and hence, in the sense of type ($).
This completes the proof of the theorem.
As a direct consequence of this theorem, we have the following
COROLLARY 1.1 (a)
In the above theorem, if{X~ }(s=1,2; ••• ) is
asymptotically equivalent (~) to another sequence, {yS }(s=1,2, ••• ), then
it holds that
(1.11)
(b)
In the above theorem, if we replace the condition (i) by
the condition (i)' {X~}(S=1,2, ••• ) converges in law to some fixed distribution,
Z say, of the continuous type,
Then it holds that
(1.12)
In the next place, we state and prove the following
THEOREM 1.2 Let {(x~,X~) }(s=1,2, ••• , )-be a sequence of twodimensional random variables satisfying the following conditions:
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(X~)(S=1,2, ••• ) has the property Cl(~) and B(~) simultan-
(i)
eously.
(X~)(S=1,2, ••• ) converges in probability to e. non-zero
(ii)
constant ;".
Then, it.holds that
and
Since the in probability convergence of ~ to ;" implies
PROOF:
that of l/~ to
1/;",
one may prove (1.13) only, and in doing this, there is
no harm in assuming that;" =1.
From the inequalities
P(~ < a) - p(~ < a ~, ~ > "I ~ p(~ ~ 0)
2
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and
I6 being any positive constant less than unity, it follows that
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(1.15)
s·
l
P(X < a) -
~
P(~ < a)
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.:s sup (x.:s
~<
I x-x' I < 21
80 /
Xl) + p(X:
6
8
~
0) + p<lX:-11 > 6)
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Since {~}(S=1,2, ••• ) has property B(,s), it can be assumed that,
for any given 6>0, there eXists a constant M(>O), such that
for all s ~ so' for some so.
assures us that there
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exis~
Moreover, the property Cl(~) of {X~}(S=1,2, ••• )
a constant 60 > 0 such that
sup
p(x ~ ~ < x') < €/4
I(x-x' 1 < 6o
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6
p(~ ~ 0) < €/4 and p(~~-ll>-£) < €/4,
2M
for all s >
S".
o
Hence if we take the value of 6 in (1.15) as 6 = 6 /2M, then it
o
holds that
(1.16)
~ < a) - P(~
p(-
sup
-co < a <
00
~
<a) 1<
€
for all s > max (s , SI, S"), which proves the theorem.
-
000
The following is straightforward from this theorem.
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COROLLARY 1.2
In the above theorem, if we replace the condition
(i) by the fo1iowing
(i)' {X~}(S=1,2, ••• ) converges in law to some probability distribution, Z
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say, of the continuous type,
Then it holds that
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(1.17)
and
(1.18)
Now, in the next place, we shall discuss another type of problem.
We often meet with the following type of problem:
Let {x(n) }(s=1,2, ••• )
and {Y(n) }(s=1,2, ••• ) be two sequences of random variables which are asymptotically equivalent ($) as S_CD, and let {(p~~p~, ••• ,p~) )(S=1,2, ••• ) be a
s
sequence of real n-vectors such that p.~l, i=l, ••• , n. We are asked whether
J.
it is true or not that
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(1.19)
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where
(1.20)
I~
s
( s
s)
s
Z(n) = Zl' ••• , Zn ' Zi
= Pis
s
Yi , i = 1, ••• ,
nj
s=1,2, ••••
The answer to this question would be 'no' in general, but, in
some special cases, (1.19) is true.
In fact, if we take the variable X~ in
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~
]
Theorem 1.2 as the unit distribution whose mass point being pS, then the theorem
states that (1.19) holds true for the case when n = 1.
This result can be
shown to be true for the n-dimensional case.
THEOREM 1.3 Let (X(n))(S=1,2, ••• ) and (Y(n))(S=~,2, ••• )be
two sequences of random variables which are asymptotically equivalent
(~)
as
s-.oo, and let (Z(n) )(S=1,2, ••• ) be the same as defined in (1.20) with the
J
J
1
J
1_
J
1
same vectors (p~, ••• , p~) given there.
Suppose that one of the original sequences, (x(n))(S=1,2, ••• )
say, has the properties Cn G$) and B(;3) simultaneously.
PROOF:
It is noted that (Y(n))(S=1,2, ••• ) has also the same
properties as (X(n))(S=1,2, ••• ), and moreover, in order to prove (1.19) it
sufficies to show the following
(1.21)
Furthermore, there is no loss of generality in assuming that
s
p.~ > 0, i = 1, ••• , n; s = 1,2, ••••
J
J
Then (1.19) holds true.
Let
€
> 0 be any given constant.
Then, the propertyB(P) of
(Y(n))(S=1,2, ••• ,) assures the existence of a positive integer So and a
]
member of p(n)' M(n) say, whose closure being compact, such that
J
(1.22)
1
I_
for all s >
f
yS
P
-
6
0
(n) (M(n)) > l-€
•
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For any given (al , ••• , an)' let us define as
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Then, since
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we have
••• , an)
... ,
~
)
an) . -< p (n)(As(n)'
/
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where we have put
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j
I:
Let us put
I~
then, this must be a disjoint sum. of at most n-members of
empty set when a , ••• , an are sufficiently large.
l
(1.23) it follows that
~(n)'
and be the
Thus, from (1. 22) and
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·1
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for all s > s •
o
Since (Y(n)}(S=1,2, ••• ) has the property Cn(~)' there exists a
positive integer s' such that
o
for all s > s' •
o
. Hence, from (1. 24) we obtain
(1.25 )
for all s > max (s , s'), which proves (1.21) and hence the theorem.
-
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0
In the final place, we note that
LEMMA 1.3
If (x{n)}(S=1,2, ••• ) and (Y(n)}(S=1,2, ••• ) are
asymptotically equivalent ($) as 8~CO, and (X(n) }(s=1,2, ••• ) converges in
probability to a point ~(n) = (~l' ••• , ~n)' then (Y(n)}(S=1,2, ••• ) converges
in probability to
~(n)
as s -*CO
•
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2.
On a class of measurable transformatioll3 preserving type (8) asymptotic
equivalence.
In this section we shall treat the following problem:
Suppose
{x{n) = (~, ••• , X:)}(S=1,2, ••• ) and {Y(n) = (Y~, ••• , Y~)}(S=1,2, ••• )
are asymptotically equivalent (,) as s400 ,and let
(2.1)
be a transformation from !R(n) to !R(l)' where z(n) = (Zl"'"
(s=1,2, ••• ,) is a sequence of real vectors.
Zn)' and {A.~, ... ,A.~)}
Put
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for each s.
Then, what conditions should be imposed in order that these two
random variables are asymptotica.lly equivalent in the sense of type ($) as
s-;oo.
An answer is given by the following theorem.
THEOREM 2.1 Under the situation stated above, suppose the
following condition is satisfied:
One of the sequences {x{n)}(S=1,2, ••• )
and {Y(n)}(S=1,2, ••• ) has the properties CN($) for any given Nand B(,9)
simultaneously. Then it holds that
1
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PROOF: Put
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Then, it is clear that there exist· a subsequence {Sl J of {sJ and a vector
(~l' ••• , ~n) such that
(2.4)
For our proof below, there is no harm in assuming that (2.4) holds for the
original sequence {sJ.
Now, put
(s = 1,2, ••• ),
and
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Then, by Theorem 4.1 of [lJ, it is seen that (2.3) is equivalent to
(2.7)
Suppose (2.7) is false.
Then, without any loss of generality,
we can assume that there exist a sequence of the members of }((l) , (E(l) }(s=1,2, ••• ),
and a positive number
~
such that
(2.8) .
For this sequence {E(1)J(S=1,2, ••• ), let us put
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Since, by the assumption of the theorem, Lemmas 1.1 and 1.2, for any given
€
> 0, there exists a member B(n) of
~(n)'
whose closure being compact, such
that both of the inequalities
hold simultaneously fo.r sufficiently large values of s.
Let us put
Then, by (2.4) and the compactness of the closure of B(n)' for
any given 6 > 0, there exist positive integers Nand N' (N<N') depending only
on .6, and a set of N' mutually disjoint members of ~( n )' (E( n.,
)·1 •••. , E()N'}
n
say, such that
(2.10)
N
I: E
k=l
C
(n)k -
AS
C
N'
I: E
(n) -k=l
(n)k
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for sufficiently large s, and
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(2.11)
,
But, since (X(n)}(S=1,2, ••• ) and hence (Y(n)}(S=1,2, ••• ) were assumed to
ha.ve the property CN(S) for any fixed N, for
€
given in (2.9) there can be
found a positive number 6o > 0 such that, for large s,
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~ .
('
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Nt
yS
P (n)( E
E
) <
k=N+l (n)k
€
Nt
and P (n)( E
E
) <
k::N+l (n)k
€
if we choose 6 in (2.11) such that 6 -< 60 •
Hence from (2.10) we have
and
for sufficiently large s.
Furthermore, from (2.9) it follows that
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and
for sufficiently large s.
Thus, by (2.12), (2.13), (2.14) and (2.15), we have
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Since 6$(n) (X(n)' Y(n» ..... o (s·~), it follows from (2.16) that
(2.17)
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...
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for sufficiently large values of s, or
(2.18)
If we choose E small, (2.18) contradicts (2.8), which proves the
theorem.
COROLLARY 2.1 If (X(n)}(S=1,2, ••• ) converges in law to some
Y(n)= (Y , ••• , Yn) of the continuou's type, then it holds that
l
I~
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eI'
s ••• , . ~s ) }(s=1,2, ••• ).
for any sequence of real vectors { (~.,
n
~
This follows from Lermna 1.1(c), Lemma 1. 2(b) and the above
theorem.
The problem we discussed in this section originates in the following
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application.
EXAMPLE:
Let Y(n) = (Yl , ... , Yn) be a random variable which is
distributed according to the n-dimensional independent normal distribution
N(O, In)' and suppose that a sequence, (x(n) = (~, ... , x:.)J(S=1,2, ••• ,),
converges in law to Y(n) as s--+co.
Furthermore, let {(~~, ... , ~~) }(s=1,2, ... )
n
be a sequence of real vectors such that i~l (~~)~co as s..-,)a>.
Then, the variables
(2.20)
n
XS = i~l
(~ + ~~)
18
2
, s
= 1,2,
.
•••
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{'
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are shown to be asymptotically equivalent (~) as. s~oo to the non-central chi~
square distribution of degrees of freedom n with the non-centrality parameter
p2 =
s
·I:n1
~=
(r..~s. )2.
In fact, this is shown as follows:
Put
then, this is distributed according to the non-central chi-square distribution
of degrees of freedom n with non-centrality parameter p2.
s
Now, since
and
the asymptotic equivalence
is equivalent to
n
n
Moreover, the two dimensional random variab1es,{(.I: ~,.I:1 r..~ Y.))(S=1,2, ••• )
~=1
~ ~=
~
~
has the property CN(~) for any given N. Hence, if it holds that
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or eq,uivalently
with IJ.{ = A.~
IPs'
i=1, ... , n, assuming that the seq,uence {(IJ.~, ••• ,. 1J.:)}(S=1,2, ... )
is convergent, then Theorem 4.4 of [lJ is applicable to show that (2.23) holds
true.
It is not so difficult, by using an analogous method to the
proof of Theorem 2.1 above, to prove (2.25).
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3. Asymptotic equivalence (13) of' marginal random variables.
f
We shall be concerned, in this section, with the f'ollowing type
of' problem:
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Le~ (X(k)' Y(J,)' Z(m)) }(S=1,2, ••• ) be a sequence of' n(=k+14m)
dimensional random variables, where k,
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and mare f'ixed independently of' s, and
let
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Hs(Z(m)) =
J Fs(Z(m) I X(k)'
Y(J)) d GS(X(k)' Y(J))
m(k)x mel)
i
be the cumulative distribution f'unction of' the marginal Z(m)' where Fs(Z(m)IX(k)'
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..
,
and G (X(k)' Y(J)) stands f'or the conditional cumulative distribution function
s
of' Z(m).g1ven (X(k)'
•
•
Y(,~))
= (X(k)' Y(J)) and the cumulative distribution
f'unction of' (X(k)' YU)), respectively•
How, suppose that the f'ollowing conditions are satisf'ied:
(i)
The marginals (X(k)}(S=1,2, ••• ) are asymptotically
equivalent (~) to some other sequence of' random variables, (X(k)}(S=1,2, ••• )say.
(ii)
For the marginals (Y(/)}(S=1,2, ••• ) there can be f'ound
a sequence of' real vectors, P\,tf)=(A.~, ••• , tj)}(S=1,2, ••• ,), such that the
sequence
,
... ,
...
)
converges in probability to the point (1, ••• , 1) as s~oo ,whereY(/)~'.'" Yj).
Let us designate the cumulative distribution f'unction of' X(k) byGs (X{k)) ,and
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put
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..
Then, clearly this is a cumulative distribution function of some m-dimensional
""S
random variable, Z(m) say.
Under the situation stated above, what additional conditions
should be imposed in order that the asymptotic equivalence
holds true?
Answers in the special cases when m = 1; k = 1,2;
1 = 0 has been
given in [2J.
In the first place, we shall state and prove the following theorem,
which is an extension of Theorems 3 and 4 of [2 J.
THEOREM 3.1
Let ((x(k)' Zem» )(s=1,2, ... ) be a sequence of
n-dimensional random variables (n = k + m) for which k and m are fixed independentlyof s, and let Hs(Z(m»' Fs(Z(m)I x(k»
and GS(x(k»
b~ the cumulative
distribution functions of Z(m)' of the conditional variable Z(m) given X(k)=X(k),
s
and of X(k)' respectively.
Suppose the following conditions are satisfied:
(i)
For the marginals (X(k) = (X~, ••• , ~»)(S=1,2, ••• ), there
exists another sequence of random variables fX(k) = (~, ••• , ~) )(s=1,2, ••• ),
which is asymptotically equivalent
(p) to (x(k»)(S=1,2, ••• ). For this sequence
(X(k) l(s=1,2, ••• ) there can be found two sequences of real vectors, (C(k) =
(c~,
••• ,
C~»)(S=1,2,
••• ) with
C~ > 0
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(U(k)=(i~,· .. ,ij~) }(s=1,2, ... >el
( i = 1, ••• , k), and
(s=1,2, ••• ) such that the sequence of random variables
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(d(k)=(d~, ••• ,d~»)
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f
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given by
u~ = ; (X~ -d~), i = 1, ••• , kj s = 1,2, •••
c
i
converges in the sense of type ($) to some fixed k-dimensional random variables
(ii)
The conditional cumulative distribution Fs(Z(m)IX(k)) is
continuous in (Z(mY' X(k)) jointly over !R(m( !R(k)"
Then, it holds that
(3.5 )
s.
.
s
s
·.....s
where Z(m) ~s the marg~nal of (X(k)' Z(m)) and Z(m) stands for a random variable
whose cumulative distribution function is given by
-
-s
Gs (X(k)) being the cumulative distribution function of X(k).
PROOF: Firstly, it is noted that the sequence of random variables
(U(k)}(S=1,2, •• ~ ) given by (3.4) has the property B(~).
proof of the theorem goes quite similarly to that of Theorem
The
In order to show (3.5), it is sufficient to prove that
sup
(3.7)
z(m)
E
/Hs(Z(m)) -
~(Z(m))
!R(m)
23
/""')0,
(s400),
because, in the case of equal basic spaces, type (~) and type (~) asymptotic
equivalence are mutually equivalent.
Since, for each s,
and
as zi ~ -
for some i or as
00
s
a point z(m)
s
= ( zl'
zi~ + 00
for all i, i = 1, ••• , m, there exists
h cthat
••• , zS)
m su
(3.8) IHs(Z(m»-Hs(Z(m»
1= SUP/Hs(Z(m»-Hs(Z(m» I,
s=1,2, ••• ,
z(m)€!R(m)
where Z(m) is allowable to take the value +
00
as some but not all of' its
components.
Since the cumulative distribution function of U(k)18 given by
for each
(3.9)
5,
it follows that
sup
f Gs(X(k» - G(X(k» /-*0, (s~oo)
x(k)€ !Jt(k)
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where x(k) = (xl' ••• , ~) and G(X(k)) stands for the cumulative distribution
function of U(k).
It also follows from the condition (i) of the theorem that
(3.10)
IGS(X(k)) - G(X(k)) I~o,
sup
(s~oo).
x(k)€ m(k)
Now, we can see that
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Since the function of the variable x(k)' Fs (Z(m) 1X(k))' is bounded
and continuous over m(k)' there exist a convergent subsequence of (Fs (Z(mf-(k)))
(s=1,2, ... ,),
(Fs,(Z(~)IX(~))}(S'~OO) say,
and a limit function <:flO(X(k)) such
that 0 ~ ~O(X(k)) ~ 1 for all x(k) in m(k)' and ~O(X(k)) is continuous over
m(k)"
Moreover, the convergence
can be regarded to be uniform on any given compact subset of m(k).
Since the sequence tUCk) }(s=1,2, ••• ), and hence U(k)' has the
property B(~), there eXists, for any given
€
> 0, a member of
whose closure being compact, such that the inequalities
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~(k)' B(k) say,
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hold simultaneously for sufficiently large values of s, where
(3.12 ) X!(k)
=
YS)
-s
1 . .8 s
Xl' ... , "k with Xi =--s(Xi-d ), i = 1, ... , k,
i
c
(-s
i
whose cumulative distribution function being GS(X(k»'
for large s,
Hence, it holds that,
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and
On the other harld, we have
and
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for sufficiently large st.
But, it is shown by using the multi-dimensional extension of the
He11y-Bray Theorem [3J, (3.9) and (3.10) we get
and
for large values of s.
we obtain
for sufficiently large values of s', which means that
sup
/Hs,(Z(m)) -
Hs,(Z(m))/~O' (s'-Ioo).
z(m) € !R(~)
Hence (3.7) should be true, which proves the theorem.
The following is a direct consequence of this theorem.
.e
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COROLLARY 3.1 Under the same situation as in the theorem,
s
suppose that the marginal Z(m) is of the continuous t;ype, i.e., is absolutely
continuous with respect to the Lebesque measure ~(m) over (!R(m)' B(m)) for
each s, wh.ose probability element being given by
where ps(Z(m)IX(k)) designates the conditional probability density function
's
given X(k) = x(k)·
Suppose the following conditions are satisfied:
(i)
{x{k)}(S=1,2, ••• ) satisfies the same condition as (i) of
the theorem.
~".
it
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el
Then, it holds that
'-s
where Z(m) stands for a variable whose probability element being
.-
ps(Z(m)lX(k)) d Gs (X(k)) d z(mf
Now, in the next place, we shall prove the following theorem.
.
S
}
THEOREM 3.2
Let {S
(Y(/)' Z(m))
(s=1,2, ••• ) be a sequence of
n-dimensional random variables (n
(i)
=..f + m), for which it is' assumed that
the conditional cumulative distribution function of Z(m)
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given Yet) = Y(f)' Fs(Z(m)\Y(~)) is continuous in (z(m)' YU?)) jointly over
m(n) for each s, and
(ii)
for the marginals {Y
ef )}(S=1,2,
••• ), there exists a
sequence of real vectors {~~, ••• , ~;)}(S=1,2, ••• ) such that ~~ > 0, i=l, ••• ,~,
and the sequence
{Ye/)
= (~/~~, ••• , YJ/~)}(S=1,2, ••• ) converges in
probability to (1, ... , 1) as s-Joo.
s
'::;s
Then, two random variables, Z(m) and Z(m)' whose cumulative
distribution functions being
where Gs(YV)) stands for the cumulative distribution function of' YW)' and
are asymptotically equivalent (IJ) as s ~oo •
For Yet) = (Yl , ... , ~), put Y(/) = (A.~ Yl ' ... , 1~).
Then the cumulative distribution function of Yet) is given by Gs (y(,) ), and
PROOF:
it becomes that
s·
For each s, there exists a point z(m)
29
such that
(3.27)
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Hs(Z(m)) - Xs(Z(m))
I = sup I Hs(Z(m)) - Hs(Z(m)) I
Z(m)E !R(m)
The left-hand member of this equality is, for any given 6 > 0,
-.
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where we have put 1(1) = (1, ... , 1) and Gs(Y(f)) = Gs(y<t)).
For each s, the function
is continuous in Y{f) and bounded over !RCf)' and consequently, the sequence
{cps(Y(.f))}(S=1,2, ... ) has a uniformly convergent subsequence on the compact
domain {Y(/)
I Y(.f(l(.f)1
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6), i.e., there exist a subsequence {Sl) (which
may depend on 6) of {s) and a lim!t function cP 0 (y(j ) ), defined over !R (p) ,
such that
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1<
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(3.29)
I tp-st(YC/))
sup
IY(.f) -1(1)/ <
- cpo(Y(R))1
~o, (s~oo).
6
It is clear that cP0 (Y(.f)) is continuous and bounded over fJ/. (/) and CPo (1 (f)) = 0,
and, for any given
€
> 0, we can choose 6 >0 such that
o
Hence, for the first member of the last expression of (3.28), we
have
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for sufficiently large values of st.
For the second member of the last expression of (3.28), it is
clear that
dG t
s
.e
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for sufficiently large values of s'.
It follows from (3.31) and (3.32) that
(3.33)
z
sup
\" H • (Z( »
e!R
s
m
(m)
(m)
-
Iis . (Z( m)}I .....o,
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(s'--.) (0).
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Hence, if we assume that
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then this leads us to a contradiction, which proves the theorem.
The following is a direct consequence of this theorem.
COROLLARY 3.2
In the above theorem, if, instead of condition
(i), we impose the following
(i)' the conditional probability density function of Z(m) given
Y(i)
= y(/)' ps(Z(m)/y(.f»
is continuous in (zCm)' YC/»
jointly over !R(n)'
for each s.
s
-s
.
Then, two random variables, Z(m) and Z(m)' whose probability
density function being respectively given by
and
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el
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(3.36)
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are asymptotically equivalent (p) as s...; 00.
In the f'inal place, we state the f'ollowing theorem without proof',
which is a easy consequence of' the above two theorems.
THEOREM 3.3 As was stated in the beginning of' this section,
let (X(k)' Y(J)' Z(m))}(S=1,2, ••• ) be a sequence of' n-dimensional random
variables (n = k +.1+ m), f'or which the f'ollowing conditions are satisf'ied:
(i)
For the second marginals, (Y(~) = (Y~, ••• , Yep))}
(s=1,2, ••• ) , there exists a sequence of' real vectors (~(/)=(~~'···' ~i)}
(s=1,2,
(Y~/~~~
) with
s
~i
> 0, i = 1, ••• ,/., such that the sequence .(-s
~(..()
=
, ~/~;)}(S=1,2, ... ) converges in probability to (1, ••• ,1) as
(ii)
S-7OO.
For the conditional distributions of' X(k) given Y(!) = ~~)
respectively f'or each s, (X(k)}(S=1,2, ••• ))there exist two sequences of' real
vectors, (c~, ... , C:)}(S=1,2, ••• ) with c~ > 0, i = 1, ••• , k, and
(d~, ... , d:)}(S=1,2, ••• ), and a sequence of' k-dimensional random variables
. lU(k) = (U~, ••• , U:) }(s=1,2,
...
), having the property B(S), such that
(X(k) = (~, ••• , ~)}(S=1,2, ••• ) and (X{k) =
(x~, ... , ~)}(S=1,2,
••• )
are asymptotically equivalent (IJ) as s~oo, where
x~
(iii)
=
c~ ~
+
d~,
i = 1, ••• , k; s = 1,2, •••
The conditional cumulative distribution f'unction of' Zem)
c
given x(k) = X(k) and Y !) = y(!), Fs(Z(m)f x(k)'
-e
y(~)) is continuous in
(Z(m)' x(k)' y(~)) jointly over ~(n)' f'or each s.
s
.....s
Then, two random variables, Z(m) and Z(m)' whose cumulative
distribution f'unctions being given respectively by
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and
. are asymptotically equivalent (13) as s....)-oo, where GS(X(k)' y(.t)) and Ks(X(k)
stand for the cumulative distribution functions of (X(k)'
(s=1,2, ••• ) given by (3.37) respectively.
Y(,P»
and rX(k»)
Hence, if Z(m)'s are absolutely
continuous, and the conditional probability density functions ps(Z(m),1 x(k)'
y(~»IS given x{k)= x(k) and y(~)= Y(I) are continuous in (Z(m)' x(k)' y(~»)
jointly over !It(n)' then, under the conditions (i) and (11) given above, Z(m)
and ~
Z(m) are asymptotically equivalent (d)
p
as s
-".lQ),
with
i=l, ••• ,m
The author is deeply grateful to Mrs. Judy Zeuner for her
nice typewriting of this manuscript.
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REFERENCES
[lJ
S. Ikeda
"On certain types of asymptotic equivalence of real
probability distributions, I.
of their properties," Univ. of North Carolina, Inst. of
Statistics Mimeo Series, 455 (1965).
design with m associate classes under the
Neyman model," Unif. of North Carolina, Inst.
of Statistics Mimeo Series, 454 (1965).
,
[3J M. Loeve
Probability Theory, Van Nost., 1955.
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Definitions and some
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