- Open Access
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.
- dependent random variable
- dominated random sequence
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.
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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