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

Abstract

For $p\ge 1$ let ${\phi }_p(x)=x^2/2$ if $|x|\le 1$ and ${\phi }_p(x)=1/p|x{|}^p-1/p+1/2$ if $|x|>1$. For a random variable $\xi$ let ${\tau }_{{\phi }_p}(\xi )$ denote $\inf \{a\ge 0:{\forall }_{\lambda \in \mathbb{R}}\ln \mathbb{E}\exp (\lambda \xi )\le {\phi }_p(a\lambda )\}$; ${\tau }_{{\phi }_p}$ is a norm in a space $Su{b}_{{\phi }_p}=\{\xi :{\tau }_{{\phi }_p}(\xi )<\infty \}$ of ${\phi }_p$-subgaussian random variables. We prove that if for a sequence $({\xi }_n)\subset Su{b}_{{\phi }_p}$ ($p>1$) there exist positive constants $c$ and $\alpha$ such that for every natural number $n$ the following inequality ${\tau }_{{\phi }_p}({\sum }_{i=1}^n{\xi }_i)\le c{n}^{1-\alpha }$ 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 ${\phi }_p$-subgaussian random variables.

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