Abstract
We use our maximum inequality for $p$-th order random variables ($p>1$) to prove a strong law of large numbers (SLLN) for sequences of $p$-th order random variables. In particular, in the case $p=2$ our result shows that $\sum f(k)/k < \infty$ is a sufficient condition for SLLN for $f$-quasi-stationary sequences. It was known that the above condition, under the additional assumption of monotonicity of $f$, implies SLLN (Erdos (1949), Gal and Koksma (1950), Gaposhkin (1977), Moricz (1977)). Besides getting rid of the monotonicity condition, the inequality enables us to extend thegeneral result to $p$-th order random variables, as well as to the case of Banach-space-valued random variables.
Highlights
N stands for the set of positive integers, N0 = N ∪ {0}
Corollary 1 Let, n ∈ N0 be a sequence of X-valued random variables such that for some 1 < p < ∞ and each k, n ∈ N0
In Gal and Koksma, 1950 it was extended to monotone sequences f (m) satisfying (2)
Summary
Given a sequence (ξn), n ∈ N0 of X-valued random variables denote a+b−1 The main objective of this note is to prove the following theorem and some of its consequences. We apply Theorem 1 to quasi-stationary sequences. Corollary 1 Let (ξn), n ∈ N0 be a sequence of X-valued random variables such that for some 1 < p < ∞ and each k, n ∈ N0
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