On the strong law of large numbers for ϕ-sub-Gaussian random variables

Abstract

UDC 517.9

For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1.$  For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote $\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$  $\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\};$ $\tau_{\varphi_p}$ is a norm in a space ${\rm Sub}_{\varphi_p}=\{\xi\colon \tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-sub-Gaussian random variables. We prove that if for a sequence $(\xi_n)\subset {\rm Sub}_{\varphi_p},$  $p>1,$ there exist positive constants $c$ and $\alpha$ such that for every natural number $n$  the inequality $\tau_{\varphi_p} \Big(\sum_{i=1}^n\xi_i \Big)\le cn^{1-\alpha}$ holds, then  $n^{-1}\sum_{i=1}^n\xi_i$ converges almost surely to zero  as $n\to\infty.$ This result is a generalization of the strong law of large numbers for independent sub-Gaussian random variables [see   R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly, 94, 295–299 (1987)] to the case of dependent $\varphi_p$-sub-Gaussian random variables.