確率添字を伴った確率過程の極限定理に関する研究

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

確率添字を伴った確率過程の極限定理に関する研究

杉万, 郁夫

https://doi.org/10.11501/3065619

出版情報：Kyushu University, 1992, 博士（理学）, 論文博士バージョン：

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Studies

on

Limit Theorems with Random Indices

Ikuo S U G IMAN

Fukuoka University

January 1993

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CONTENTS

1. Introduction

2. Uniform independence on essential parts

3. Uniform c-independence

(

D-independence

)

4. Anscombe condition

5. Examples

Acknowledgement

References

1

6

11

14

19

23

24

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1. Introduction

We shall consider the following problem that is frequently called the limit theorem of the randomly indexed sequence of random variables or the limit theorem with random indices or, simply, the random limit theorem.

Let

(0, 0, P)

be a probability space and

{Yn;

n

� 1}

be a sequence of random variables defined on

(0, 0, P)

. Suppose that

(1)

in distribution as n --+ oo , where F is a distribution function. And let

{ rn;

n

� 1}

be a sequence of random indices, that is, positive integer valued random variables defined on the same probability space

(0, 0, P)

that

{Yn;

n

� 1}

is defined on. Suppose that

(2)

1n some sense as n --+ oo

.

We consider when can we conclude that the randomly indexed sequence of random variables

(3)

in distribution as n --+ oo

.

Note that the limiting distribution function of

{ Yrn;

n

� 1}

is the same one of

{Yn;

ⁿ

� 1} .

In the simplest case

Yn = ( 1/fo) Lk=l Xk (

ⁿ

� 1),

where

{Xn;

ⁿ

� 1}

is a sequence of independent and identically distributed random variables with :finite variances, this problem was treated :firstly by Robbins

[25)

and discussed fully by Tate

[31).

In their papers, they required the sequences of random indices

{ rn;

n

� 1}

to be independent of

{Yn;

n

� 1} .

^This

assumption has been frequently treated from then till now

([22),[24]).

The :first elementary result of the limit theorems with random indices seems to be the following theorem.

(6)

THEOREM

1.1 Let {Yn;

n

� 1} be a sequence of random variables such that {Yn;

n

� 1} converges zn distribution to a distribution function

F

as

n ---+ oo .

And let { rn;

n

� 1} be a sequence of random indices that is independent of {Yn;

n

� 1} and diverges in probability to infinity as

n ---+ oo

. Then the randomly indexed sequences {Yrn;

n

� 1} of random vari ables also converges in distribution to

F

as

n ---+ oo .

While, by Anscombe

[6]

and D.R.Cox, it was shown that a statistical estimation formula valid for fixed sample sizes remained valid when the sample size was determined by a sequential stopping rule in certain sequential estimation procedure. Their results suggested that most of all fixed-sample-size formulae might be valid generally for sequential sampling if sample size was large.

But, of course, the condition in the previous theorem that a sequence of random indices

{ rn;

n

� 1}

is independent of

{Yn;

n

� 1}

is not applicable to these problems in sequential estimation, because the sequence of random indices is a sequence of stopping times dependent of

{Yn;

n

� 1}

on these problems.

The following theorem is the first essential result given by Anscombe

[5]

in the case when nothing is supposed concerning the dependence of the sequence of random indices

{

Tn; n

� 1}

THEOREM

1.2 Let {Yn;

n

� 1} be a sequence of random variables such that {Yn;

n

� 1} converges zn

distribution to a random variable

Y

as

n ---+ oo

( Yn

==> F

) . And let { rn;

n

� 1} be a

sequence of random indices such that

Tn

/

n

converges to 1 in probability as

n

---+

oo .

(7)

Then in order that

Yrn

converges in distribution to

Y

as

n --+ oo

(

Yrn ==? F

) , it is sufficient that for any c >

0 ,

there exists o >

0

such that

limsup

n-+oo P( max i;li-nl�6n I Yi

^-

Yn I > c)

<

c

^·

(4)

This condition of uniform continuity in probability is now frequently called the Anscombe condition and is applicable to the simplest case of the central limit theorem of a sequence of normalized sums of independent and identically distributed random variables with finite variances. Then this condition is very useful on the above problems and has been applied to these problems and others by many authors

([8],[9],[12],[13],[18],[19],[33]).

The purposes of most of all papers in this area have been to give the generalized versions of this condition and theorem 1.2 and extend the class of limit theorems that these generalized versions of this result are applicable.

It is known at present that for the generalized verswns of the Anscombe condition to hold, it is required that

{Yn;

n

� 1}

satisfies a kind of generalized Kolmogorov inequality.

Moreover if it is required some condition relative to the order of convergence of

rn/

an where

{ a n

; n

� 1}

is an increasing sequence of positive numbers tending to infinity as n --+ oo , for example, an equals to the integer part of the median of ^Tn, some generalised version of the Anscombe condition is applicable to the central limit theorem for a general class of the sequences of dependent random variables in which the weakly stationary sequences having the convergent sums of covariances are included as a familiar case.

Such limit theorems are useful in various kinds of applications, for example, sequential analysis, queueing theory, reliability theory, renewal theory and so on

([21]).

(8)

Anscom be has also conjectured in his paper [5], the necessity of his condition if no further condition is imposed on { rn;

n

2: 1} , and many authors has given the necessity in the case when (1) is a kind of the central limit theorem of a sequence of normalized sums of independent random variables. Aldous (3] has given the first general result for the necessity of the Anscom be condition in the case when { Yn;

n

2: 1} is a stationary sequence of random elements of a separable metric space. His result was extended to nonstationary case and so on ([11 ],[27]).

The necessity of the condition, not only gives the rightness of condition, but also clarifies the difference between these conditions. And the limit theorems (I) are characterized by the class of sequences of random indices that satisfies the limit theorems (3) of the randomly

indexed sequences of random variables, for exa

mp

le

,

the mixing limit theorem, the stable

limit theorem and so on ([4],[23],[29]).

Dorea et al. [14] has proposed the uniform independence condition of the sequence of random indices (Def.3 .1 ), (D-independence called in the present thesis) and the idea of an approximation of random indices as a fully stochastic version of the Anscombe condition.

This approach vehicles for extending the applicability of the independence condition through approximation of a sequence of random indices by other sequence of random indices that satisfies some kind of independence condition and suggests the importance of the independence conditions or more generalized conditions.

The only one result that the limit theorem with the modified D-independence property are

characterized by some class of sequences of random indices satisfying the limit theorems with

random indices has been given by Sugiman [29].

(9)

In the present paper, we introduce the condition of uniform independence on essential parts of the sequences of random indices and give the main result in the present thesis in which the limit theorems with the uniform independence property introduced in the present thesis are characterized by the class of sequences of random indices with the same essential parts satisfying some measurability condition

(

_Section

2).

These condition and result are clearly generalizations of the D-independence condition and the result that characterize the D-independence condition

(

_Section

3).

Also we show in Section 4 that these are generalizations of the Anscombe condition as another sufficient condition for the limit theorem with random indices and the results by Aldous and so on.

Furthemore we give an example of the limit theorem of the randomly indexed sequence that only our condition is applicable and show that this condition is also valuable as the sufficient condition and extends the class of the limit theorems of the randomly indexed sequences of random variables with the independence condition that the idea of an approximation of random indices is applicable

(

_Section

5).

(10)

2. Uniform independence on essential parts

Let Nk denote the lattice of positive integer k-dimensional points where k is a positive integer. The points in N

k

are denoted by

m = (m1, m2, ... , mk) ,

^J1

= (n1, n2, ... , nk)

and so on. The partial order in N

k

is denoted by

m t

¹¹ if and only if

mi � ni

for each i

=

1

,

2

,

... ,k.

Let

{Y!!:.;

¹¹

E

Nk

}

be a set of random elements of a metric space

(S, d)

with the Borel a--field 3 converging weakly to a random element

Y ( Y!!.

^===}

Y

as ^{J1 � oo}

)

of

(S, d)

that defined on the same probability space

(0,

^0,

P)

that

{Y!!:.;

¹¹E Nk

}

is defined on , that is, for any ^{c; >}

0

and any

P(Y E

^·

)

-continuous set

B E

3

,

there exists

N E

Nk such that

Il

E

U(N)

implies that

I P(Y!!:.

E

B)- P(Y

E

B) I

< ^£

,

where

U(J1)

is the set of successors of

IlE

Nk , that is

U(rr) ={mE Nk;m t rr}.

Let also

{ r!!:.;

¹¹

E

Nk

}

be a set of Nk-valued random indices defined on the same probability space

(0,

^0,

P)

that

{Y!!.;Il E

Nk

}

and

Y

are defined on such that

;!!.

diverges to infinity in probability

( ;!!:. ----.

^oo as ^{11 � oo}

)

, that is, for any ^{c; >}

0

^{and any}

M E

Nk , there exists

N E

Nk such that J1

E U(N)

implies that

P(r!!. E U(M))

> 1-c; •

Let

{ D!!:.;

¹¹

E Nk}

be a set of subsets of Nk . Then

{ D!!:.;

¹¹

E Nk}

is said to be an essential part of the set

{ r !!.;

^J1

E

Nk

}

of random indices, if for any ^{c; >}

0

, there exists

N E

Nk such that J1

E U(N)

implies that

P( T!!. E DJ

^>1 -c; •

(5)

Now let us introduce the uniform independence condition mentioned in Section 1 which is similar to the essential c:-independence

([30]).

(11)

Let

{

^0!bm;^n_,

m E Nk}

be a set of sub-cr-algebra.s of 0 with double indices. And let

{D!!.;n. E Nk}

be a set of subsets of

Nk .

Since

r!!.

diverges to infinity in probability as

n.---+

^oo , there exists a subset

{L!!.;n. E Nk}

^of

Nk

which satisfies that

L!!:..---+

^oo ^as

n.---+

^oo , that is, for any

L E Nk

there exists N

E Nk

such that

n. E

^U

(

N

)

^implies

that

L!!. t L_

, and for any � > 0 there exists N

E Nk

such that

n. E

^U

(

N

)

^implies

that

P(r!!. t L�

^{> 1-} ^{� . If}

{D!!.;n. E Nk}

is an essential part of

{ r!!.;n. E Nk}

^{, let}

D� = D!!.

ⁿ

U(L�

^{for each}

n. E Nk

^{, then}

{ D�; n. E Nk}

is also an essential part of

{r!!.;n_ E Nk}

^. Thus without loss of generality we can assume that the above mentioned set

{ D!!.; n. E Nk}

of subsets of

Nk

satisfies that for any

L E Nk

there exists N

E Nk

^such

that

rr E

^U

(

N

)

implies that

D!!.

^c

U(L_ ) .

� > 0.

DEFINITION 2.1

A doubly indexed set

{

0!b!r!:.;n_,

mE Nk}

of sub-a--algebras of 0 is said to be

uniformly

�-independent

of

( {Y!!.;rr E Nk} , {D.!!..;rr E Nk})

^for

BE

^3,if for any set of mappings

m!!. ; Nk ---+ D!!.,

^{that is,}

m!!.(E ) E D!!.

for all n_ ,

p_ E Nk

, and any set of countable partitions

{A!!:.,t;p_ E Nk}

such that for any

n. E Nk, { A!!Jt;p_ E Nk}

is a 0!!Jm-measurable countable partition for some

m E Nk ,

there exists N

E Nk

such that

rr E

^U

(

N

)

implies that

(6)

Throughout the present thesis, we denote

¢(A, B)= P(A

ⁿ

B ) - P(A )P(B ) .

A set

{

^0!!Jm;^n_,

m E Nk}

of sub-a--algebras of 0 with double indices is said to be uniformly �-independent of

( { Y!!.;rr E Nk} , {D!!.;rr E Nk} )

^{, if}

{

0!b!r!:.;n_,

m E Nk}

^is

uniformly �-independent of

( {Y.!!..;n. E Nk}, {D.!!..;n. E Nk})

^{for any}

P(Y E

^·

)

-continuous

(12)

set B

E

³

.

And

{ 0!bm; Jl, m E Nk}

is said to be, simply, uniformly independent of

( {Y!!.;JlE Nk}, {D!!.;JlE Nk}), if {0!bm.;rr,m E Nk}

is uniformly £-independent of

( {Y!!.;JlE Nk}, {D!!.;JlE Nk})

for any ^c> 0.

Remark 2.2

The summation in

(6)

always absolutely converges since

{A!b�;E_ E Nk}

is a countable partition of 0 _{for each}

Jl E Nk .

The following theorem shows that the limit theorem having the uniform independence property on some essential parts stated in the previous definition are characterized by the class of sequences of random indices with the same essential parts satisfying some measurability condition.

THEOREM 2.3

It is necessar y and sufficient that a doubly indexed set { 0!bm; Jl, m E Nk} of sub-u-algebras of 0 is uniformly independent of ( {Y!!.;Jl E Nk} , {D!!.;Jl E Nk} ) , in order that the limit theorem of randomly indexed set of random elements

holds as

J1---+ oo

) for any set of random indices { r!!.;

J1

E Nk} such that (

a

) For any Jl E Nk ,

r!!:.

is e!b!I!.- measurable for some m E Nk , and (

b

) { D!!.; Jl E Nk} is an essential part of { r !!.; Jl E Nk} .

Proof.

(7)

(

Sufficiency

)

For any c > 0 _and B

E

³, there exists N1

E Nk

such that for Jl

E U(!!Jj

(13)

and

}£

^E

D'!l

I P(Y�

E

B) - P(Y

E

B) I

< €

(8)

s1nce

{ D'!l;

n. E

Nk}

satisfies that for any

L_

^E

Nk

there exists

N

E

Nk

such that ^{n. E}

U(N)

implies that

D'!l

C

U(L)

. We define the set of random indices

on

{

r

'!l

E

D'!l}

otherwise

(9)

where

L'!l

is any point of

D'!l

for each n. E

Nk

, then (a) implies that

v'!l

satisfies that

v'!l

E

D'!l

and is 8!bm-measurable for some

m

E

Nk

. Hence it follows from

(9)

and (b) that there exists

N2

E

Nk

such that n. E

U(!!.Jj

implies that

I P(Yr!!.

E

B) - P(Yv!!.

E

B) I

(

10₎

< €

and if ^{n. E}

U(!!.J)

, then the uniform independence implies that

I P(Yv!!.

E

B)- P(Y

E

B) I

� I � !s.ED!!. _¢( {Y�

E

B}, {v!!.

=

l£}

)

I

(11) +

� /s.ED!!.I P(Y�

E

B)- P(Y

E

B) I

^·

P(v!!.

=

l£)

< 2 _{€ .}

Thus the sufficiency is proved from

(

10) and (11).

(Necessity) Suppose the contrary. There exist

B

E 3 _, _€> 0 , the set

{ m'!l;

11 E

Nk}

of mappings from

Nk

to

D'!l

, the set of countable partitions

{

^{A!b�; E.}E

Nk}

such that for any ^{11 E}

Nk

,

{

^A!b�;

_E

E

Nk}

is a e!bm-measurable countable partition for some

m

E

Nk

and the countable subset

{ n'}

of

Nk ,

(12)

(14)

We define

v!!.

=

m!!.(lf)

on A!bk ^,then

{ v!!.;

11 E

Nk}

satisfies (a) and (b), but

(13)

This implies that

{Y11!!.;

11 E

Nk}

does not converge weakly to

Y

, since

{ D!!.;

11 E

Nk}

satisfies that for any L. E

Nk

there exists

N

E

Nk

such that 1l E

U(N)

implies that

Thus the proof is complete. o

(15)

3.

Uniform c-independence (D-independence)

The following definition is the most applicable one in the conditions on independence between

{Yn;

n 2:

1}

and

{ rn;

n 2:

1}

in the case k =

1 .

DEFINITION

3.1 ([14])

A sequence

{ rn;

n 2:

1}

of random indices is said to be

D-in dependent

of a sequence

{Yn;

ⁿ 2:

1}

of random variables if for any ^{c >} 0 and any continuity point ^x of a distribution function F , there exist M, N 2:

1

such that

(14)

for any sequence of indices

{ mi; i

2:

1}

such that

mi

2: M for all

i 2: 1

and any countable partition

{Ai;i

2:

1}

E U

n?:_N IT(rn)

where

IT(rn)

is the family of all Tn-measurable countable partitions.

To characterize this D-independence condition by some class of sequences of random indices that are D-independent of

{Yn;

n

2: 1}

, we modify the above condition to be a condition for only the sequence

{Yn;

ⁿ2:

1}

of random variables.

Let

{ 0 n;

11 E

Nk}

be a set of sub-a-algebras of

0 .

DEFINITION

3.2 ([29])

{ 0 n;

^{11 E}

Nk}

is said to be

D-independent

of a set

{Yn;

^{11 E}

Nk}

of random elements of

(

^S,

d)

, if for any c > 0 and any

P(Y

E ·

)

-continuous set B _E3 , there exist M, N _E

Nk

such that

(15)

for any set of mappings

m(

^·

)

from

Nk

to U

(

M

)

(16)

{AE.;E. E Nk} E

U r!EU(!:!J II(0J where II(0.J is the family of all 0.!!.-measurable countable partitions.

Now we define two classes of sets of random indices. Let r+(

{0!!.;n_ E Nk})

be the class of all sets of random indices which satisfies

(a) T.!!. ^{----+ oo} as

n_---+

oo

,

and

(b) for ^rr

E Nk

' there exists

m E U(rr)

such that T

!!.

IS em-measurable.

COROLLARY 3.3

([29])

In order that

(16)

holds as

rr ---+ oo

for all

_{{ T !!:.;}rr

E Nk} E

r+ ({ 0!!:.; ^rr

E Nk}) , it is necessary and sufficient that the set

_{0n;n.

E Nk} of sub-a-algebras of

₀

is D-independent of the set {Y!!.;n_ E Nk}

of random elements of (S, ^d)

^.

Proof.

In order that {0?1:.; ^rr

E Nk}

is D-independent of

{Y!!.; n_ E Nk} ,

it is necessary and sufficient that for any ^€ ^> 0 _{and any}

P(Y E

· )-continuous set B

E

2 , there exists

N E Nk

such that

(17)

for any set of mappings

m(

·

)

_from

Nk

_to

U(N)

{AE.;E. E Nk}

which is a 0?1:.- measurable countable partition for some ^rr

E U(N)

_. Then we set

{ er!:.+m-1

;

rr,

m E Nk}

_and

{ U(n_); 11 E Nk}

_as

{

_0!!J!!!;

11, m E Nk}

_and

{ D!!:.; 11 E Nk}

in definition

2.1,

_where 1 =

(1, 1,

· · ·,

1) E Nk

. This D-independence condition is equivalent that {0n,mi!?!.,

mE Nk}

is uniformly independent of

( {Y!!.;11 E Nk} , {D!!.;11 E Nk} ) ,

(17)

since ^{n' E} U(nJ implies that D!l!... CD!!. and

{0!G!!!;

^mE

Nk}

^c

{0.!!..!!!;

^mE

Nk} .

Thus the proof is followed by theorem

2.3.

^o

We define another class of sets of random indices. Let

T( { 0!!.; Jl

^E

Nk})

be the class of all sets of random indices which satisfies (a) and

The following result is obvious from the previous corollary, since

T( { 0!!.;

^{n. E}

Nk})

1s a

COROLLARY

3.4 ([14])

If the set {0n;Jl

^E

Nk} of sub-(}-a/gebras of0 is D-independent of the set {Yn;Jl

^E

Nk}

- -

of random elements of (S, d) , then

(18)

4.

Anscombe condition

In this section, we consider the special case

k

⁼

1

to compare with the classical conditions and the results by Aldous and so on. Moreover we assume the metric space

(

S,

d)

is separable, then

d(Yi, Yj)

is measurable for each i,

j 2: 1 ,

and this assumption is required in the proof of proposition

4.3.

The result that we show as corollary of theorem

2.3

in this section is the result relative to the generalized version of the Anscombe condition to the nonstationary case stated in introduction.

Let

{ kn;

ⁿ

2: 1}

be a non decreasing sequence of positive numbers that represents the norming sequence in the limit theorem of

{Yn;

ⁿ

2: 1}

, for example,

kn

⁼

fo

or

kn

⁼^a

fo

in the simplest central limit theorem of the normed sums of independent and identically distributed random variables with finite variances ^a2 stated in introduction. And let

{an;

ⁿ

2: 1}

be a nondecreasing sequence of positive integers tending to infinity as n --+ oo that represents an approximation sequence of sequences of random indices in some sense, for example,

an

equals to the integer part of the median of

rn .

DEFINITION

4.1 ([11])

{Yn;

ⁿ

2: 1}

is said to satisfy the generalized A nscombe condition

(

to apply to the limit theorems of the nonstationary sequences of random variables

)

, if for any ^{c >}0 there exists

8

^>0 such that

(19)

Let

{ 8n;

ⁿ

2: 1}

be also a nonincreasing sequence of positive numbers tending to 0, that

(19)

represents the order of convergence of

krn/ kan

in probability to a positive constant or a positive random variable and so on.

DEFINITION

4.2 ([27])

{Yn; n 2: 1}

is said to satisfy

the refined Anscombe condition (

by the order of convergence

(20)

is degenerate as

n

---jo oo .

Now we define the classes of sequences of random indices according as the above conditions.

Let

T

⁼

T( { kn;

n

2: 1}, {an; n 2: 1})

be the class of all sequences of random indices such that

(21)

in probability as

n

---jo oo .

And let

T({8n;n 2: 1})

⁼

T({kn;n 2: 1},{an;n

^>

1};{8n;n

^>

1})

be the class of all sequences of random indices such that

(22)

as n---jooo .

Firstly, we shall treat the refined Anscom be condition and the class

T( { 8n; n 2: 1})

^of

sequences of random indices, and show that the uniform independence for some

{ Dn; n 2: 1}

is a generalization of this version of the Anscombe condition.

(20)

PROPOSITION 4.3

The following three statements are equivalent.

(

^a

) {Yn;n 2: 1} satisfies the refined Anscombe condition A({8n;n 2: 1}) .

(

^b

) {0n,m; n, m 2: 1} is uniformly independent of ( {Yn; n 2: 1} , {Dn; n 2: 1 } ) , where

(

^c

) Yrn

^==? F

holds as n--+

^oo

for all {rn;n 2: 1}

^E

T({8n;n 2: 1}) .

Proof.

((

^a

)

=>

(

^b

))

^Since

lim n-+oo kn =lim n-+oo an=

^oo and

lim n-+oo On=

0 , then

lim n-+oo min Dn =

^{oo .}

Let ^c > 0 and

B = Br(P) = {x

^E S;

d (x

,

P) � r}

be

B± = Br±e(P)

and

B

are

P(Y

^E^·

)

-continuous. Let

{ mn; n 2: 1}

be a sequence of mappings from N to

Dn

and a

sequence of countable partitions

{ { An,k; k 2: 1} ; n 2: 1 } .

Then

Ef=1 P( {Ymn(k)

^E

B } n An,k)

� P( ( Uf=1( {Ymn(k)

^E

B } n An,k) ) n Nne)+ P(Nn)

� Ef=1 P( {Ymn(k)

^E

B} n An,k n { d(Ymn(k), Yan)

^{< c}

}) + P(Nn) (23)

� Ef=1 P( {Yan

^E

B+} n An,k) + P(Nn)

= P(Yan

^E

B+) + P(Nn)

where

Nn = { max iEDn d(Yi, Yan) 2:

^c

} .

This implies that

limsup n-+oo E�1 P( {Ymn(k)

^E

B} n An,k)

^<

P(Y

^E

B+) +

^{c .}

(24)

Similarly the result that

(25)

is obtained. This implies that

liminf n-+oo Ef=1 P( {Ymn(k)

^E

B} n An,k)

>

P(Y

^E

B-)-

^c

. (26)

(21)

It follows from

lim n-+oo min Dn

^{= oo} that there exists a positive integer N such that for any

n

^{> N} and k >

1 I P(Ymn(k)

^E

B) - P(Y

^E

B) I

^{< €}

which implies that

I �k::1 P(Ymn(k)

^E

B) · P(An,k) - P(Y

^E

B) I

^{< €}^•

Combining this inequality with

(24)

and

(26),

we have

limsup n-+oo I �k=l ¢( {Ymn(k)

^E

B}, An,k) I

:::; limsup n-+oo I �k=l P( {Ymn(k)

^E

B}

ⁿ

An,k) - P(Y

^E

B) I + limsup n-+oo I �k=l P(Ymn(k)

^E

B)· P(An,k)- P(Y

^E

B) I

<

P(B+- B-)+ 2c

� 0 as

c

� 0

.

Then

(

b

)

holds.

(27)

(28)

(29)

((

b

)

^=?

(

c

))

^If

{rn;n � 1}

^E

T({8n;n � 1}),

then

{Dn;n � 1}

mentioned in

(

b

)

^is

obviously essential parts of

{ Tn; n � 1}

. Therefore the proof is complete from the sufficiency of theorem

2.3. ((

c

)

^=?

(

a

))

^Since

(S, d)

is separable, this is the same result as theorem

2.2

in

[24].

Thus the proof is omitted.

Thus the proof is complete.o

The following corollary is well known result relative to the generalized Anscombe condition and the class

T

of sequences of random indices. But, we shall prove this result as the union of the above refined results.

COROLLARY

4.4 In order that

(30)

(22)

holds as n

---+ oo

for all { Tn; n � 1}

^E

T

,

it is necessary and sufficient that { Yn; n � 1}

satisfies the generalized

A

nscombe condition.

Proof.

The sufficiency can be proved similarly to the classical results, then we shall only prove the necessity. Suppose the contrary. Then there exist ^{c >}

0

and a strictly increasing sequence of positive integer

{ nm; m � 1}

such that

(31)

for all

m � 1

, where

Dm

=

{i � 1 ; I ki- kanm I� kanm/m}

^. Then if we put

6n

=

1/m

for all

nm-1

+

1 � n � nm

, then

{Yn; n � 1}

does not satisfy the condition

A({8n;n � 1})

. But this contradicts that

(30)

holds for all

{rn;n � 1}

^E

T

, since

T( { 8n; n � 1})

is a subclass of

T

. Thus the proof is complete.o

(23)

5. Examples

The first example is a weakly convergent sequence of random variables

{Yn; n � 1}

which does not satisfy any generalized version of the Anscombe condition and a sequence of random indices

{rn;n � 1}

which is D-independent of

{Yn;n � 1}

and satisfies

(21).

Example

5.1 ([14])

Let

(0, 0,P)

=

( (0,1]

, B n

(0,1]

^,

JL)

_where B is the family of Borel sets and ^JL is the Lebesgue measure restricted on

(0,1] .

If

n

⁼

2p

, then

Yn

⁼

{:

on ^on

^{(0, 1/2]} (1/2, 1].

(32)

And if

n

⁼

2p - 1 ,

then

Yn

⁼

{ ^�

^onon

(1/2, 1] . ^{(0, 1/2]} (33) {Yn; n � 1}

is obviously weakly convergent and does not satisfy any generalized version of the Anscombe condition.

Define

on

(1/2, 1/2

+

1/n]

(34)

otherwise .

Then

{ Tn; n � 1}

is D-independent of

{Yn; n � 1}

, since if

mi � 1

for all i

� 1

and

{Ai;i � 1}

is nontrivial (there exists i

�

1 such that

Ai

# n and

Ai

# ¢>) partition of n which belongs to

Un;:::1II( Tn)

where

II( Tn)

is a family of all

Tn-

measurable countable

partitions,

I ��1 </>( {Ym; � X}, Ai) I

=

I </>( {Ym;1 �X}, Ai1)

+

</>( {Ymi2 � X}, AiJ I

<

4/N

(35)

where

Ai1 , Ai2

are the only two subsets of n in this partition not equal to n and ¢> . And

{rn;n � 1}

is obviously satisfy

(21)

with

an= n .

(24)

Conversely, the following example is a weakly convergent sequence of random variables

{Yn; n 2: 1}

which satisfies the classical Anscom be condition and a sequence of random indices

{rn;n 2: 1}

which satisfies

(21)

with

an=

ⁿ and is not D-independent of

{Yn;n 2: 1}.

Example

5.2. ([30])

Let

(

_{0, 0,P}

)

=

( [0,1)

, B ⁿ

[0,1) , 1-l)

_where _B is the family of Borel sets and ^1-l is the Lebesgue measure restricted on

[0,1) .

If

n

⁼

�;=o 2P

+

m ( 0 � m � 2q+

¹

- 1 )

_{, then}

(36)

otherwise .

{Yn;n 2: 1}

is obviously degenerate to

1(w) = 1.

And if

0

< ^c <

1 , 0

< 8 <

112

and

2q+l

< _-

n

⁼

�q- 2P

_p-0 +

m

<

2q+2

' then

P( max k�l

^;1

k-nl

�

on I Yk - Yn I 2:

^€

)

This implies that

{ Yn; n 2: 1}

satisfies the classical Anscom be condition with

an = n . (37)

Define

T n

⁼

n

+

m

on

[ ( m - 1) I Vn , m I Vn )

_for

1 � m � Vn- 1

and

Tn

⁼

n

+

Vn

+ ^s

s =

0. {rn; n 2: 1}

obviously satisfies

(21)

with

an= n.

But, for any

N,

_M

2: 1

^{, let} q

2: 1

such that

2q-l 2: N

_and

2q 2:

M

.

Let k ⁼

4q (2: M)

and Ai ⁼

{ Tk

⁼k +

i}

for

i 2: 1

, then

{

Ai;

i 2: 1}

is a

Tk-

measurable countable partition.

Let

ni

⁼

�;:�2P

+

i- 1 (2: N)

_for

i 2: 1

, then Ai ⁼

{Yni

⁼

0}

⁼

[ (i- 1)12q , il2q) (1 � i � 2q-l)

. It follows from

(38)

(25)

that

I E�l ¢( {Yn,

_<

1/2}, Ai) I

� E;:ll ¢( {Yn,

_<

1/2}, Ai) - I E�2q ¢( {Yna

_<

1/2}, Ai) I

� (1 - 2-q)2 - 2 E�2q P(Ai)

=

(1 - 2-q)2 - 2-(q-l) .

Then

{ Tn; n � 1}

is not D-independent of

{Yn; n � 1} .

(39)

Even in the case k ⁼

1

, theorem

2.3

contains many results for the limit theorem of the randomly indexed sequences in Section

3

and Section 4.

On the following example of the limit theorem of the randomly indexed sequence of random variables, the sequence of random indices

{ Tn; n � 1}

is not D-independent of the sequence of random variables

{Yn; n � 1} .

Of course, it does not satisfy any generalized version of

the Anscombe condition. But the sequence of random indices is uniformly independent of

( {Yn;n � 1},{Dn;n � 1})

for some essential part

{Dn;n � 1}

_of

{rn;n � 1}.

_{Thus the}

example shows that the uniform independence property, not only unifies the D-independence property and the Anscombe condition, but also has more applicable limit theorems as the sufficient condition.

Example

5.3. ([30])

Let

(0, 0,P)

=

( [0,1)

, B n

[0,1) ,

_1-L

)

where B is the family of Borel sets and 1-L is the Lebesgue measure restricted on

[0,1) .

If

n

₌_k

(mod 3) ,

_then

on

[ k/3 , (k+l)/3 )

(40)

otherwise .

{Yn; n � 1}

is obviously weakly convergent and does not satisfy any generalized version of the Anscombe condition.

(26)

And define

{ ^n-k

Tn =

n-k + 1

on

[ 1/6 , 1/2 ]

(41)

otherwise .

Then

{rn; n 2: 1}

obviously satisfies

(21)

with

an

=

n .

And for any N,

M 2: 1 ,

let

n1=3( [M/3]+1 ) (2:M), n2=n1+2 , ni=M (i2:3)

and

A1=[1/6,1/2], A2=

A! , Ai = ¢ ( i 2: 3)

^. Then

I ��1 ¢( {Yni

<

1/2}, Ai) I

=I¢( [1/3, 1], [1/6, 1/2] ) + ¢( [o, 2/3], [1/6, 1/2]c ) 1 (42)

= 1/6.

Thus

{rn;n2:1}

is notD-independentof

{Yn;n2:1}.

But,let

0n,m={¢ ,[1/6,1/2],[1/6,1/2]c,o}

for

n,m2:1

and

Dn={n-k,n-k+1}

for

n 2: 1

, then

¢( {Y3p

<

x }, [1/6, 1/2]) = ¢( {Y3p-2

<

x }, [1/6, 1/2]) = -1/18

and

¢({Y3p

<

x},[1/6,1/2]c) = ¢({Y3p-2

<

x},[1/6,1/2]c) = 1/18

for all p

2:1 ,

0 <

x

<

1.

Thus

{0n,m; n, m 2: 1}

is un iformly independent of

( {Yn; n 2: 1} , {Dn; n 2: 1} )

^.

(27)

Acknowledgement

I am very greatful to Proffessor N .Furukawa of Kyushu University for his helpful comments and many valuable suggestions, not only during the preparation of the present thesis, but also through all of my investigation.

I also wish to thank Proffessor E.Isogai of Niigata University for his useful ad vices, and Proffessor M.Watanabe and Proffessor Y.Ebihara of Fukuoka University for their continual hearty encouragements.

(28)

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(33)