Suppose we have a simple random walk: $X_n\pm1$ with equal probabilities.
For any finite $n$, $E[\sum_{k=1}^nX_k]=0$. Does it imply that $E[\lim_{n\rightarrow \infty}\sum_{k=1}^nX_k]=0?$
Thank's!
Edit:
We know that $\sum_{k=1}^nX_k$ diverges. Formally, $\limsup\sum_{k=1}^nX_k=\infty$ and $\liminf\sum_{k=1}^nX_k=-\infty$ almost surely. So, probably this observation may help.
Edit 2:
Let $S_n=\sum_{k=1}^nX_k$ Following the discussion below, $\lim_{n\rightarrow\infty }S_n$ does not exist. So, the expectation does not make sense. To put it formally, we don't have pointwise convergence because $\limsup S_n(\omega)\ne \liminf S_n(\omega)$ almost surely. Hence, $\lim S_n(\omega)$ does not exist a.s. Is it correct?