an:06151183
Zbl 1274.60098
Nane, Erkan; Xiao, Yimin; Zeleke, Aklilu
A strong law of large numbers with applications to self-similar stable processes
EN
Acta Sci. Math. 76, No. 3-4, 697-711 (2010).
00292219
2010
j
60F15
strong law of large numbers; moment inequality; self-similar processes; stable processes
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 [\textit{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.
Zbl 1112.60024