Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that for any such sequence
$\Pr (S_n\geq 0 \; \mbox{ i.o. ( infinitely often}))>0$
Any ideas? relevant references? If necessary, additional assumptions on the variance may be added, e.g. that there is a bound $B$ such that $Var(X_i)\leq B$.
Thanks