Let $(a_n)_{n\in\mathbb{N}}$ a sequence of $\mathbb{R}$ which converges to $0$ and $(X_i)_{i\in\mathbb{N}}$ a sequence of iid variables.
Assuming that the logarithmic moment of the $X_i$ exists, does the serie $\sum_{i\geq 0}|a_i X_i|$ converges almost everywhere?