The question is related to this question.
Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ converges a.s. to some non degenerate $S$ and
$$\lim_{n\rightarrow\infty}E[\max(S_n,0)]=\infty$$
Is there exists a condition under which $\lim_{n\rightarrow\infty}E[\max(S_n,0)]=E[\lim_{n\rightarrow\infty}\max(S_n,0)]$? $\max(S_n,0)$ is not monotonic sequence so we cannot use monotone convergence theorem. Also, the reverse Fatou's lemma cannot be applied since $S_n$ is not bounded.
Thank's!