A strong law of large numbers with applications to self-similar stable processes

Abstract

Let p ∈(0,∞) be a constant and let ξn ⊂ Lp(Σ,F,ℙ) be a sequence of random variables. For any integers m, n ≥ 0, denote \(S_{m,n}=\sum{_{k=m}^{m+n-1}}\xi_k\). It is proved that, if there exist a nondecreasing function ϕ ℝ+ → ℝ+ (which satisfies a mild regularity condition) and an appropriately chosen integer a ≥ 2 such that $$\sum_{n=0}^\infty\begin{array}{c}\sup\mathbb{E}\\ {k \geq 0}\end{array}|\frac{S_{{k,a}^n}}{\varphi (a^n)}|<\infty$$ then $$\lim_{n \rightarrow \infty}\frac{S_{0,n}}{\varphi(n)}=0$$ This extends Theorem 1 in Chobanyan, Levental and Salehi [3] and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.