Consider a random variable $X$ that is Gaussian with mean $\mu$ and variance $\sigma^2$. I am trying to understand whether the following statement holds:
\begin{equation} \frac{X}{n} \xrightarrow[n\rightarrow\infty]{\text{a.s.}} 0. \end{equation} I feel that this statement should not hold for almost sure convergence but it should hold for in probability convergence. I used Markov inequality and could not prove the result for almost sure convergence, because an assumption of the form $\mathbb{E} [ |X|] < \infty$ was needed. Any help would be really appreciated! Thank you for your time!