On almost sure limiting behavior of a dependent random sequence
© Fan and Wang; licensee Springer 2013
Received: 28 August 2012
Accepted: 3 January 2013
Published: 18 January 2013
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.
In reference , Chow and Teicher gave a limit theorem of almost certain summability of i.i.d. random variables as follows.
Theorem (Chow et al., 1971)
- (3)is non-decreasing for , then i.i.d. are summable, i.e.,
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 )
Lemma 1 (Chow et al., 1978 )
Let be an () martingale difference sequence, if , then a.c. converges.
2 Strong law of large numbers
is non-decreasing for .
Let , then is a martingale difference sequence.
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.
Proof Note that is a sequence of m-dependent random variables, then , Corollary 4 follows directly from Corollary 3. □
Definition 2 (Stout, 1974)
holds for any integer .
Thus, using the Kroneker lemma, Theorem 2 follows. □
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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