zbMATH — the first resource for mathematics

A strong law of large numbers with applications to self-similar stable processes. (English) Zbl 1274.60098
Summary: Let \(p\in (0,\infty)\) be a constant and let \(\{\xi_ n\}\cup L^p(\Omega,\mathcal F,\mathbb P)\) be a sequence of random variables. For any integers \(m, n \geq 0\), denote \(S_ {m,n}=\sum^{m+n-1}_ {k=m} \xi_ k\). It is proved that, if there exist a nondecreasing function \(\varphi :\mathbb R_ +\to\mathbb R_ +\) (which satisfies a mild regularity condition) and an appropriately chosen integer \(a \geq 2\) such that \[ \sum^\infty_ {n=0}\sup_ {k\geq 0}\mathbb E \left| \frac{S_ {k,a^n}} {\varphi(a^n)}\right| ^p < \infty, \] then \[ \lim_ {n\to \infty} \frac{S_ {0,n}}{\varphi(n)} = 0 \quad\text{a.s.} \] This extends Theorem 1 in [S. Chobanyan et al., Electron. Commun. Probab. 10, 218–222 (2005; Zbl 1112.60024)] and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

60F15 Strong limit theorems
Full Text: arXiv