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On almost sure limiting behavior of a dependent random sequence
Journal of Inequalities and Applications volume 2013, Article number: 25 (2013)
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 , be a positive non-increasing function and , , , where
-
(1)
;
-
(2)
;
-
(3)
is non-decreasing for , then i.i.d. are summable, i.e.,
for some choice of centering constants , if and only if
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 ℕ denote the set of positive integers, be a stochastic sequence defined on the probability space , i.e., the sequence of σ-fields in ℱ is increasing in n, and are adapted to random variables , denotes the trivial σ field and the indicator function.
We begin by introducing some terminology and lemmas.
Definition 1 (Adler et al., 1987 [2])
Let be a sequence of random variables, and it is said to be stochastically dominated by a random variable X (we write ) if there exists a constant , for almost every , such that
Lemma 1 (Chow et al., 1978 [3])
Let be an () martingale difference sequence, if , then a.c. converges.
Lemma 2 Let be a sequence of random variables. If , then for all ,
Proof By the integral equality
it follows that
□
2 Strong law of large numbers
In this section, we always assume that , is a positive non-increasing function and , , , where
-
(1)
;
-
(2)
;
-
(3)
is non-decreasing for .
Theorem 1 Let be a sequence of random variables with . If , then
Proof To prove (2.1) by applying the Kronecker lemma, it suffices to show that
Since , (1) guarantees that . Choose such that implies
whence for entailing
Put , obviously, . Note that and the condition , we have
which shows
Let , then is a martingale difference sequence.
Since
Note that
which implies that a.c. Hence, by Lemma 1, we have 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 be a sequence of random variables with . If for some , , then
Corollary 2 Let be a sequence of random variables with and for some ,
where , , , if for all large ,
then
Corollary 3 Let be a sequence of random variables with . Further, let and , . If , then for any ,
Proof Since is an adapted stochastic sequence and , by Theorem 1, we have for that
Therefore, we have
□
Corollary 4 Let be a sequence of m-dependent random variables. Further, let and . If there exists a random variable X such that and , then
Proof Note that is a sequence of m-dependent random variables, then , Corollary 4 follows directly from Corollary 3. □
Definition 2 (Stout, 1974)
Let be a sequence of random variables, and let . We say that the sequence is *-mixing if there exists a positive integer M and a non-decreasing function defined on integers with such that for , and , the relation
holds for any integer .
It has been proved (cf. [4]) that the *-mixing condition is equivalent to the condition
for and implies
Theorem 2 Let be a sequence of *-mixing random variables with . Further, let and , . If , then
Proof By Corollary 3, we have, for each ,
Since is *-mixing, by (2.8) and (2.9), we obtain
Thus, using the Kroneker lemma, Theorem 2 follows. □
References
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
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
Chow YS, Teicher H: Probability Theory. Springer, New York; 1978.
Stout WF: Almost Sure Convergence. Academic Press, San Diego; 1974.
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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The authors declare that they have no competing interests.
Authors’ contributions
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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DOI: https://doi.org/10.1186/1029-242X-2013-25