  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.
