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

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 $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge 2$ such that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}} {φ(a^n)} \bigg|^p < \infty,$$ Then $$\underset{n\to \infty }{\lim }\frac{S_{0,n}}{\phi (n)}=0a.s.$$ 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.

arXiv comment9 pages