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.
Highlights
1 Introduction In reference [ ], Chow and Teicher gave a limit theorem of almost certain summability of i.i.d. random variables as follows
Throughout this paper, let N denote the set of positive integers, {X, Xn, Fn, n ∈ N} be a stochastic sequence defined on the probability space (, F, P), i.e., the sequence of σ fields {Fn, n ∈ N} in F is increasing in n, and {Fn} are adapted to random variables {Xn}, F denotes the trivial σ field {, } and [·] the indicator function
We say that the sequence {Xn, n ∈ N} is *-mixing if there exists a positive integer M and a non-decreasing function φ(n) defined on integers n ≥ M with limn φ(n) = such that for n > M, A ∈ F m and B ∈ Fm∞+n, the relation
Summary
Theorem (Chow et al, ) Let a(x), x > be a positive non-increasing function and an = a(n), An = Throughout this paper, let N denote the set of positive integers, {X, Xn, Fn, n ∈ N} be a stochastic sequence defined on the probability space ( , F , P), i.e., the sequence of σ fields {Fn, n ∈ N} in F is increasing in n, and {Fn} are adapted to random variables {Xn}, F denotes the trivial σ field { , } and [·] the indicator function. Lemma (Chow et al, [ ]) Let {Xn, Fn, n ∈ N} be an Lp ( ≤ p ≤ ) martingale difference sequence, if
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