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On almost sure limiting behavior of a dependent random sequence

Abstract

We study some sufficient conditions for the almost certain convergence of averages of arbitrarily dependent random variables by certain summability methods. As corollaries, we generalized some known results.

MSC:60F15.

1 Introduction

In reference [1], Chow and Teicher gave a limit theorem of almost certain summability of i.i.d. random variables as follows.

Theorem (Chow et al., 1971)

Let a(x), x>0 be a positive non-increasing function and a n =a(n), A n = k = 1 n a k , b n = A n / a n , where

  1. (1)

    A n ;

  2. (2)

    0< lim inf n b n n a(log b n ) lim sup n b n n a(log b n )<;

  3. (3)

    xa( log + x) is non-decreasing for x>0, then i.i.d. {X, X n } are a n summable, i.e.,

    T n = A n 1 k = 1 n a k X k C n 0a.c.

for some choice of centering constants C n , if and only if

E|X|a ( log + | X | ) <.

Motivated by Chow and Teicher’s idea, in this paper we consider the problem of arbitrarily dependent random variables and their limiting behavior from a new perspective.

Throughout this paper, let denote the set of positive integers, {X, X n , F n ,nN} be a stochastic sequence defined on the probability space (Ω,F,P), i.e., the sequence of σ-fields { F n ,nN} in is increasing in n, and { F n } are adapted to random variables { X n }, F 0 denotes the trivial σ field {Φ,Ω} and 1 [ ] the indicator function.

We begin by introducing some terminology and lemmas.

Definition 1 (Adler et al., 1987 [2])

Let { X n ,nN} be a sequence of random variables, and it is said to be stochastically dominated by a random variable X (we write { X n ,nN}X) if there exists a constant C>0, for almost every ωΩ, such that

sup n N P { | X n | > t } CP { | X | > t } for all t>0.

Lemma 1 (Chow et al., 1978 [3])

Let { X n , F n ,nN} be an L p (1p2) martingale difference sequence, if n = 1 E( | X n | p | F n 1 )<, then n = 1 X n a.c. converges.

Lemma 2 Let {X, X n ,nN} be a sequence of random variables. If { X n }X, then for all t>0,

E | X n | 2 1 [ | X n | t ] C [ t 2 P ( | X | > t ) + E X 2 1 [ | X | t ] ] .

Proof By the integral equality

2 0 t sP ( | X n | > s ) ds= t 2 P ( | X n | > t ) +E | X n | 2 1 [ | X n | t ] ,

it follows that

 □

2 Strong law of large numbers

In this section, we always assume that a(x), x>0 is a positive non-increasing function and a n =a(n), A n = k = 1 n a k , b n = A n / a n , where

  1. (1)

    A n ;

  2. (2)

    0< lim inf n b n n a(log b n ) lim sup n b n n a(log b n )<;

  3. (3)

    xa( log + x) is non-decreasing for x>0.

Theorem 1 Let {X, X n } be a sequence of random variables with { X n }X. If E|X|a( log + |X|)<, then

lim n 1 A n k = 1 n a k [ X k E ( X k 1 [ | X k | b k ] | F k 1 ) ] =0, a.c.
(2.1)

Proof To prove (2.1) by applying the Kronecker lemma, it suffices to show that

the series  n = 1 X n E ( X n 1 [ | X n | b n ] | F n 1 ) b n converges a.c.

Since 0<a(x), (1) guarantees that b n . Choose m 0 such that n m 0 implies

αn b n a(log b n )βn
(2.2)

whence b n αn [ a ( log b m ) ] 1 for nm m 0 entailing

k = m b k 2 a 2 ( log b m ) α 2 m .
(2.3)

Put Y n = X n 1 [ | X n | b n ] , Z n = X n 1 [ | X n | > b n ] , obviously, X n = Y n + Z n ,nN. Note that { X n }X and the condition E|X|a( log + |X|)<, we have

(2.4)

which shows

P( X n Z n ,i.o.)=0.
(2.5)

Let W n = Y n b n E( Y n b n | F n 1 ), then ( W n , F n ,nN) is a martingale difference sequence.

Since

(2.6)

Note that

E [ n = 1 E ( W n 2 | F n 1 ) ] E [ n = 1 E ( Y n 2 b n 2 | F n 1 ) ] = n = 1 E Y n 2 b n 2 < ,
(2.7)

which implies that n = 1 E( W n 2 | F n 1 )< a.c. Hence, by Lemma 1, we have n = 1 W n a.c. convergence.

Theorem 1 follows from (2.5) and (2.7). □

Theorem 1 also includes some particular cases of means, we can establish the following.

Corollary 1 Let {X, X n ,nN} be a sequence of random variables with { X n }X. If for some ε>0, E | X | log | X | 1 [ | X | > ε ] <, then

lim n 1 log n k = 1 n [ X k E ( X k 1 [ | X k | k log k ] | F k 1 ) k ] =0,a.c.

Corollary 2 Let {X, X n ,nN} be a sequence of random variables with { X n }X and for some k2,

a n = [ n ( log n ) ( log k 1 n ) ] 1 ,

where log 1 n=logn, log k n=log( log k 1 n), k2, if for all large C>0,

E | X | 1 [ | X | > C ] ( log | X | ) ( log k | X | ) <,

then

lim n 1 A n k = 1 n a k [ X k E ( X k 1 [ | X k | b k ] | F k 1 ) ] =0,a.c.

Corollary 3 Let {X, X n ,nN} be a sequence of random variables with { X n }X. Further, let F n =σ( X 1 ,, X n ) and F n ={ϕ,Ω}, n0. If E|X|a( log + |X|)<, then for any m1,

lim n 1 A n k = 1 n a k [ X k E ( X k 1 [ | X k | b k ] | F k m ) ] =0, a.c.
(2.8)

Proof Since { X n m + l , F n m + l ,n1} is an adapted stochastic sequence and { X n m + l }X, by Theorem 1, we have for l=0,1,,m1 that

n = 1 X n m + l E ( X n m + l 1 [ | X n m + l | b n m + l ] | F ( n 1 ) m + l ) b n m + l converges a.c.

Therefore, we have

 □

Corollary 4 Let { X n ,nN} be a sequence of m-dependent random variables. Further, let F n =σ( X 1 ,, X n ) and F n ={ϕ,Ω},n0. If there exists a random variable X such that { X n }X and E|X|a( log + |X|)<, then

lim n 1 A n k = 1 n a k [ X k E ( X k 1 [ | X k | b k ] ) ] =0,a.c.

Proof Note that { X n ,nN} is a sequence of m-dependent random variables, then E( X n | F n m )=E X n , Corollary 4 follows directly from Corollary 3. □

Definition 2 (Stout, 1974)

Let { X n ,nN} be a sequence of random variables, and let F n m =σ( X n ,, X m ). We say that the sequence { X n ,nN} is *-mixing if there exists a positive integer M and a non-decreasing function φ(n) defined on integers nM with lim n φ(n)=0 such that for n>M, A F 0 m and B F m + n , the relation

| P ( A B ) P ( A ) P ( B ) | φ(n)P(A)P(B)

holds for any integer m1.

It has been proved (cf. [4]) that the *-mixing condition is equivalent to the condition

| P ( B | F 0 m ) P ( B ) | φ(n)P(B),a.c.

for B F m + n and m1 implies

| E ( X n + m | F 0 m ) E X n + m | φ(n)E| X n + m |,a.c.
(2.9)

Theorem 2 Let {X, X n ,nN} be a sequence of *-mixing random variables with { X n }X. Further, let F n =σ( X 1 ,, X n ) and F n ={ϕ,Ω}, n0. If max{E|X|,E|X|a( log + |X|)}<, then

lim n 1 A n k = 1 n a k [ X k E X k 1 [ | X k | b k ] ]=0,a.c.

Proof By Corollary 3, we have, for each m1,

lim n 1 A n k = 1 n a k [ X k E ( X k 1 [ | X k | b k ] | F k m ) ] =0,a.c.

Since { X n ,nN} is *-mixing, by (2.8) and (2.9), we obtain

Thus, using the Kroneker lemma, Theorem 2 follows. □

References

  1. Chow YS, Teicher H: Almost certain summability of independent, identically distributed random variables. Ann. Math. Stat. 1971, 42(1):401–404. 10.1214/aoms/1177693533

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  2. Adler Y, Rosasky A: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl. 1987, 5(1):1–16. 10.1080/07362998708809104

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  3. Chow YS, Teicher H: Probability Theory. Springer, New York; 1978.

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  4. Stout WF: Almost Sure Convergence. Academic Press, San Diego; 1974.

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Acknowledgements

Foundation item: The National Nature Science Foundation of China (No. 11071104), Foundation of Anhui Educational Committee (KJ2012B117) and Graduate Innovation Fund of AnHui University of Technology (D2011025).

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Correspondence to Ai-hua Fan.

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WZ and FA carried out the design of the study and performed the analysis, WZ drafted the manuscript. All authors read and approved the final manuscript.

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Fan, Ah., Wang, Zz. On almost sure limiting behavior of a dependent random sequence. J Inequal Appl 2013, 25 (2013). https://doi.org/10.1186/1029-242X-2013-25

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Keywords

  • dependent random variable
  • summability
  • dominated random sequence