76
A CLASS OF STRONG LIMITS THEOREMS FOR ARBITRARY STOCHASTIC SEQUENCE
Xiaojun Cheng1,*, Weiguo Yang2 & Xiang Li 3
1, 2, 3
Faculty of science, Jiangsu University, Zhenjiang 212013, P.R.China
*Tel: +86-18796001402, E-mail address: [email protected] ABSTRACT
By use of truncation methods of random variables and Doob’s martingale convergence theorem, this paper presents a class of strong limit theorems for an adapted sequence of random variables. Some convergence theorems for martingale difference sequence and a class of strong limit theorems for independent random variables are the particular cases of the results of this paper.
Keywords: Strong limit theorem; Truncation method; Martingale difference sequence; Independent random variable.
2010 Mathematics Subject Classification: 60F15,
1. INTRODUCTION
Let{X F nn, n, 0}be a stochastic sequence on the probability space( , , ) F P , that is, the sequence of -fields { ,F nn 0}inFis increasing in n (that isFn ), andXnis measurableFn.
Almost sure behavior of partial sums of random variables has enjoyed a resurgence of research activity. A number of researchers were interested in convergence theorem of partial sums of random variables and obtained lots of classical results for sequences of independent random variables and martingale difference sequence (see [1, p.121]and[2,p.17]).
In recent years, some work has been done on the strong limit theorems for arbitrary sequences. In Ref. [3], Jardas et al. have proved a strong law of large numbers for independent random variable sequences, which generalized Chung's classical strong law of large numbers for sequence of independent random variables (see [1], p.124). In Ref.[4], Liu and Yang have proved two strong limit theorems for arbitrary stochastic sequences which extended and improved Chow's strong law of large numbers for martingale difference sequences(see[2, p.35], [5, p.66]and[10]) as well as Chung's classical strong law of large numbers for sequence of independent random variables. Yang (see[6])lately established two more general strong limit theorems in 2007,which generalized a result by Jardas et al.
for sequence of independent random variables and the results by Liu and Yang [4] for arbitrary stochastic sequences in 2003.
In this paper, we establish a class of new strong limit theorems for stochastic sequences. Yang’s strong law of large numbers for arbitrary stochastic sequences, Loeve’s convergence theorem (see [7]) and Petrov’s (see [8]) strong law of large numbers for sequences of independent random are the particular cases of this paper. In addition, the main theorems of this paper extend the main results by Jardas et al. (see [3]) and Yang (see [4, 6]), respectively.
We first give a lemma.
Lemma 1. (Doob, see [2, p.33]) Let{X F nn, n, 0}be a martingale difference sequence. Then
1 n
n k k
S
X converges . .a e on the set{
k1E X[ k2 Fk1] }. 2. MAIN RESULTSTheorem 1. Let{X F nn, n, 0}be a stochastic sequence defined as before, and{ ,a nn 1}be a
sequence of non-zero positive random variables such thatanis measurableFn1. Letn:RRbe Borel functions and letn0,n1 2,Kn1and Mn1(nN)be constants satisfying
1 2 1 2 1 2
1 2
1 2
( ) ( )
(i) 0 ,
( ) ( )
n n
n n
n n
n n
n n
x x x x
x x K and M
x x
x x
（1）
77
2
2 1 1
( )
(ii) : | ,
( ) ( ) ( )
n n
n n
n n n n n n n
A K E X F
a a X
（2）2
2 1 1
( )
(iii) : | ,
( ) ( ) ( )
n n
n n n
n n n n n n n
B K M E X F
a a X
（3）Set
1
1
1, sup 2, sup
n n
n n
K
K
and 1
1
1, sup 2, sup
n n
n n
M
M
（4）
Then
1
. . .
n
n n
X converges a e on A B a
（5）Proof. Letn0,X*nX I Xn ( n an), Let
k
be a positive integral number, and letZn 2 2
( ) ( ( ) ( ) ( ))
n Xn n an n an n Xn
, set
1
1
: | ,
k n n n
n
A K E Z F k
（6）1
1
1
inf : 1, | ,
n
k i i n
i
n n K E Z F k
（7）
Wherek , if the set on the right-hand side of (7) is empty. It is easy to see that
1
k n
m m
m
K Z
1 n
k m m
m
I m K Z
. SinceI
km
is measurableFm1, we have
1
1 1 1
1 1
1 1
|
|
| . (8)
k
k
n n n
m m k m m m k m m
m m m
n
m k m m
m n
m m m
m
E K Z E I m K Z E K E I m Z F
E K I m E Z F
E K E Z F k
SinceAk
k
, we have by (8), for all n
1 1 1 1
1 1
(9)
k
k k
n n n n
m A m m k m k m m k m m
m m m m
n n
k m m m m
m m
K Z dP K E I A Z E I A K Z E I K Z
E I K Z E K Z k
Hence we have
1 k
n
m A m
m
K Z dP k
（10）By (1), we have 1 Xnn ann Knn(Xn) n(an), as Xn an. Hence
*
1 1 1
1
1
( ( ))
( ) (1 ) ( )
( ) ( ) ( )
(1 )
n
n
n n n n
n n
k
n
k n n
n n X a n X a n
n n n n n
n
n X a n n n n n n
n n n
n A
P A X X dP X dP
a
X K X
K dP
a X a
K K Z dP Rk
（11）
78
WhereRbe a positive integral number, and Its value is determined by.By (11) and Borel- Cantelli lemma, we haveP A X( k( nX*n) . .)i o 0. Hence we have
*
1
( n n) / n . . k.
n
X X a converges a e on A
Since k
k
AA , we have
*
1
( n n) / n . . .
n
X X a converges a e on A
（12）By (1), we have Xn an Xn2n an2n Xnn ann Mnn(Xn) n(an), as Xn an. Hence
*
1 1
[ n| n ] n n n n n | n
E X F a E X a I X a F
| 1n n
n n
n n
n n n
n n
X X
E I X a F
a a
1 1( ) ( )
( ) ( ) |
n n n n
n n n n n n
n n n n n n
X X
E M M K I X a F
a K a
2 2
2 1 1
( )
(1 ) |
( ) ( ) ( )
n n
n n n
n n n n n n n
M K E X F
a a X
（13）By (13)、(3) and (4), we have
* 1
1
[ n| n ] n . . .
n
E X F a converges a e on B
（14）LetY0 0and
Yn
Xn*E X[ n*|Fn1]
an, n 1. (15) It is clear that{ ,Y F nn n, 0}is a martingale difference sequences. Observe that
2 * 2 * 2
1 1 1
* 2
1
[ | ] { [( ) | ] ( [ | ]) }
[( ) | ] . .
n n n n n n n
n n n
E Y F E X F E X F a
E X F a e
a
(16)
By (1), we have Xn an Xnn ann Mnn(Xn) n(an), if Xn an. Then
2 1 * 2 1 22
11 1 1
[ n | n ] [( n) | n ] n n n | n
n n n n n
X X
E Y F E F E I X a F
a a
11
n n |
n n n
i n n
X X
E I X a F
a a
11
( ) ( )
( ) ( ) |
n n n n
n n n n n n
n n n n n n
X X
E M M K I X a F
a K a
11
( ) (1 ) ( )
( ) ( ) ( ) |
n n n n n
n n n n n n
n n n n n n n n n
X K X
E M M K I X a F
a K a K X
11
( ) (1 ) ( )
( ) ( ) ( ) |
n n n n n
n n n n n n
n n n n n n n n n
X K X
E M M K I X a F
a K a K X
2 2
2 1 1
( )
(1 ) | . .
( ) ( ) ( )
n n
n n n
n n n n n n n
M K E X F a e B
a a X
(17) By lemma 1,
* * 1
1 1
[ | ] . . .
n n n n n
n n
Y X E X F a converges a e on B
(18)(5) follows from (12), (14) and (18). The proof of this theorem is completed.
79
From theorem 1, we can easily obtain strong limit theorem as follows:
Corollary 1. Let{X F nn, n, 0},{ ,a nn 1},{n( ),x n1}, nandnbe given as in Theorem 1, and let1Kn , 1Mn be constants satisfying
1 2 1 2 1 2
1 2
1 2
( ) ( )
0 ,
( ) ( )
n n
n n
n n
n n
n n
x x x x
x x K and M
x x
x x
Set
2
2 1 1
( )
: |
( ) ( ) ( )
n n
n
n n n n n n n
C E X F
a a X
Then1
1
. . .
n n
n
a X converges a e on C
Proof. Letting1Kn and1Mn in Theorem 1, we haveA B C, this corollary follows.
This corollary extends some strong limit theorems as follows:
Corollary 2. Let{X F nn, n, 0}and{ ,a nn 1}be given as in Theorem 1. Letn:RRbe Borel functions and let
0, 1
n n
, 1Ln and1Hn be constants satisfying
1 2 1 2 1 2
1 2
1 2
( ) ( )
0 ( ) ( )
n n
n n
n n
n n
n n
x x x x
x x L and H
x x
x x
Set
1
1
( )
: |
( )
n n
n
n n n
D E X F
a
Then
1
1
. . .
n n
n
a X converges a e on D
Proof. Letn( )x n2( )x ,n2 n, n2n, LnKn2andHn Mn2for all n in Corollary1. SinceDC, this corollary follows.
Remark 1. If1Ln and1Hn , the corollary 2 is the particular cases of Yang (see [6]). From corollary 3 we can easily obtain Yang’s strong limit theorems for a class of stochastic sequences and Loeve’s strong law of large numbers for sequences of independent random variables as follows:
Corollary 3(see [4]). Let{X F nn, n, 0}, { ,a nn 1}be given as in Corollary2. Let fn:RRbe a sequence of nonnegative, even functions such that f xn( )andx f xn( )are non-decreasing as x increases. Suppose that
1 1
( )
: |
( )
n n
n
n n n
f X
E E F
f a
.Then we have
1
1
. . .
n n
n
a X converges a e on E
Proof. Letn( )x f xn( ),n0, n1,Ln 1andHn 1for all n in Corollary2. Since DE, this corollary follows.
Corollary 4(Loeve, see [7]). Let{X nn, 1}be a sequence of independent random variables. If there exist rn(0, 1]
for alln such that
1 | n|rn
n E X
, then
n1Xnconverges . .a e .Proof. Letan 1and f xn( ) xrnfor alln andFn(X X1, 2,Xn) (F0{ , })in Corollary3. Since
80
{X nn, 1}be a sequence of independent random variables, we haveE X F[ n n1]
n . . EX a e
.This corollary follows from Corollary 3.
Corollary 5 (Petrov, see [8]). Let{X nn, 1}be a sequence of independent random variables, and let{ ,a nn 1}be defined as in corollary 2. If0pn1for alln such that
1
n n
p p
n n
n E X a
, then
k1ak1Xk converges. . a e .
Theorem 2. Let{X F nn, n, 0}be a stochastic sequence defined as before, and{ ,a nn 1}be a
sequence of non-zero positive random variables such thatanis measurableFn1. Letn:RRbe Borel functions and letn1 2,n1,Kn1and Mn1(nN)be constants satisfying
1 2 1 2 1 2
1 2
1 2
( ) ( )
(i) 0 ,
( ) ( )
n n
n n
n n
n n
n n
x x x x
x x K and M
x x
x x
(19)
2
2 1 1
( )
(ii) : | ,
( ) ( ) ( )
n n
n n
n n n n n n n
A K E X F
a a X
（20）2
2 1 1
( )
(iii) : | ,
( ) ( ) ( )
n n
n n n
n n n n n n n
B K M E X F
a a X
（21）Set
1
1
1, sup 2, sup
n n
n n
K
K
and 1
1
1, sup 2, sup
n n
n n
M
M
（23）
Then
1
1
[ (| | ]
. . .
n n n n n
n n
X E X I X a F
converges a e on A B a
（24）Proof. TakingXn*X I Xn ( n an), for alln0. LetAk,kandZnbe defined as in the proof of Theorem1.
Using a similar argument, we also prove (10) holds. By (19), we have 1 Xnn ann
( ) ( )
n n n n n
K X a
, as Xn an. Hence
*
1 1 1
1
1
( ( ))
( ) (1 ) ( )
( ) ( ) ( )
(1 )
n
n
n n n n
n n
k
n
k n n
n n X a n X a n
n n n n n
n
n X a n n n n n n
n n n
n A
P A X X dP X dP
a
X K X
K dP
a X a
K K Z dP Rk
（25）
By (25), we similarly have (12) holds. Secondly, using a similar argument to derive (18), we also prove the following conclusions by (19) and (21)
* * 1
1
[ | ] . . .
n n n n
n
X E X F a converges a e on B
(26)(24) follows from (12) and (26). The proof of this theorem is completed.
It is easy to see that Corollary 1 is a generalization of Corollary 2. The following example shows that it is really a generalization.
Example 1. Let
X nn, 1
be a sequence of independent random variables andXnhas the distribution density as follows81
2 2
2
1 1
, ,
( ) ( )
0 , 1 .
n
nx x n
f x
x n
We will show that 1
1
n n
n E X
and
1 2 1 2
1
n n
n
E X n n X
. In fact, since2 1
1 2
1 .
( )
n n
n E X x dx
n nx
Then we have 1
1
n n
n E X
. Observe that
1 2 1 2
1 2 1 2 21
( ) ( )
n n n
E X n n X x dx
n nx nx
5 2 1 2 1 2 3 2
1 1
n dx
n n x x
5 2 1 2 3 2 3 2
1 1 2
.
n dx
n x n
Then, we have
1 2 1 2
3 21 1
2
n n
n n
E X n n X
n
.11. REFERENCES
[1]. Chung.K.L, A Course in Probability Theory, seconded, New York: Academic Press, 1974.
[2]. Hall.P, Heyde.C.C, Martingale Limit Theory and Its Application, New York: Academic Press, 1980.
[3]. Jardas.C and Pecaric.J, A note on Chung’s strong law of large numbers, J. Math. Anal. Appl, 1988, (217):
328-334.
[4]. Liu.W, Yang.W.G, A class of strong limit theorems for the sequences of arbitrary random variables, Statist.
Probab, Letts. 2003, (64): 121-131.
[5]. Stout.W.F, Almost Sure Convergence, New York: Academic Press, 1974.
[6]. Yang.W.G, Strong limit theorems for arbitrary stochastic sequences, J. Math. Anal. Appl. 2007, (326): 1444- 1451.
[7]. Chow, Y. S. and Teicher, H. Probability Theory: Independence, Interchangeability, Martingales, Springer Texts in Statistics, Springer, New York, NY, USA, 2nd edition, 1988.
[8]. Petrov, V.V., Sums of Independent Random Variables, New York, Springer- Verlag, 1975.
[9]. Shiryayev.A.N, Probability, Seconded, New York: Springer- Verlag, 1996.
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