A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes

TL;DR

This paper establishes a strong law of large numbers for self-similar processes with stationary increments, providing conditions under which normalized sums of random variables converge almost surely, extending previous results in the field.

Contribution

It introduces a generalized strong law of large numbers applicable to self-similar stable processes, broadening the scope of prior theorems with a new convergence criterion.

Findings

Provides almost sure convergence criteria for normalized sums of random variables.

Extends previous theorems to a wider class of self-similar processes.

Applicable to stable processes with stationary increments.

Abstract

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if there exist a nondecreasing function $\varphi: \R_+\to \R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge 2$ such that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}} {\varphi(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty} \frac{S_{0, n}} {\varphi(n)} = 0\qquad \hbox{a.s.} $$ This extends Theorem 1 in Levental, Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.