On the strong law of large numbers for $\varphi$-subgaussian random variables

Abstract

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 $\inf\{a\ge 0:\;\forall_{\lambda\in\mathbb{R}}\; \ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\}$; $\tau_{\varphi_p}$ is a norm in a space $Sub_{\varphi_p}=\{\xi:\;\tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-subgaussian random variables. We prove that if for a sequence $(\xi_n)\subset Sub_{\varphi_p}$ ($p>1$) there exist positive constants $c$ and $\alpha$ such that for every natural number $n$ the following inequality $\tau_{\varphi_p}(\sum_{i=1}^n\xi_i)\le cn^{1-\alpha}$ 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 SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $\varphi_p$-subgaussian random variables.