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On almost sure limiting behavior of a dependent random sequence

Journal of Inequalities and Applications20132013:25

https://doi.org/10.1186/1029-242X-2013-25

  • Received: 28 August 2012
  • Accepted: 3 January 2013
  • Published:

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.

Keywords

  • dependent random variable
  • summability
  • dominated random sequence

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)
    x a ( 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 0 a.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 , n N } be a stochastic sequence defined on the probability space ( Ω , F , P ) , i.e., the sequence of σ-fields { F n , n N } 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 , n N } be a sequence of random variables, and it is said to be stochastically dominated by a random variable X (we write { X n , n N } X ) if there exists a constant C > 0 , for almost every ω Ω , such that
sup n N P { | X n | > t } C P { | X | > t } for all  t > 0 .

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

Let { X n , F n , n N } be an L p ( 1 p 2 ) 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 , n N } 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 s P ( | X n | > s ) d s = 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)

    x a ( 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
α n b n a ( log b n ) β n
(2.2)
whence b n α n [ a ( log b m ) ] 1 for n m 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 , n N . 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 , n N ) 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 , n N } 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 , n N } be a sequence of random variables with { X n } X and for some k 2 ,
a n = [ n ( log n ) ( log k 1 n ) ] 1 ,
where log 1 n = log n , log k n = log ( log k 1 n ) , k 2 , 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 , n N } be a sequence of random variables with { X n } X . Further, let F n = σ ( X 1 , , X n ) and F n = { ϕ , Ω } , n 0 . If E | X | a ( log + | X | ) < , then for any m 1 ,
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 , n 1 } is an adapted stochastic sequence and { X n m + l } X , by Theorem 1, we have for l = 0 , 1 , , m 1 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 , n N } be a sequence of m-dependent random variables. Further, let F n = σ ( X 1 , , X n ) and F n = { ϕ , Ω } , n 0 . 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 , n N } 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 , n N } be a sequence of random variables, and let F n m = σ ( X n , , X m ) . We say that the sequence { X n , n N } is *-mixing if there exists a positive integer M and a non-decreasing function φ ( n ) defined on integers n M 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 m 1 .

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 m 1 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 , n N } 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 } , n 0 . 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 m 1 ,
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 , n N } is *-mixing, by (2.8) and (2.9), we obtain

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

Declarations

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).

Authors’ Affiliations

(1)
School of Mathematics & Physics, AnHui University of Technology, Ma’anshan, 243002, People’s Republic of China

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/1177693533MATHMathSciNetView ArticleGoogle Scholar
  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. 10.1080/07362998708809104MATHView ArticleGoogle Scholar
  3. Chow YS, Teicher H: Probability Theory. Springer, New York; 1978.MATHView ArticleGoogle Scholar
  4. Stout WF: Almost Sure Convergence. Academic Press, San Diego; 1974.MATHGoogle Scholar

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