How to prove the convergence of a series for a weighted random variables?

Question

Let $\{X_n,n\ge 1\}$ be a sequence of independently identically distributed random variables with the same mean $\mu$, $a_n, n\ge 1$ be real numbers. If $\sum_{n=1}^{\infty}|a_n|<\infty$ then the random series $\sum_{n=1}^{\infty}a_n X_n$ is convergent a.s.. How to prove its inverse proposition, that is, how to prove that if the random series $\sum_{n=1}^{\infty}a_n X_n$ is convergent a.s. then $\sum_{n=1}^{\infty}|a_n|<\infty$?

Answer 1

It is not true. If $X_n = \pm 1$, each with probability $1/2$, then it is well known that $\sum_n \frac1n X_n$ converges a.s. (Special case of Kolmogorov three series theorem.)