I am trying to prove the following question:
Let $X_i$ be sequence of indepedent non-negative random variables with $E(X_i) = \infty$ and $X_i$ having different distributions.
Is it true that $$ \frac{X_1 + \cdots + X_n}{n} \to \infty \ a.s.? $$ I know that this is true when $X_i$ are i.i.d., but I have no idea how to proceed with $X_i$ having different distributions.
Any hint would be appreciated. Thank you!