I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$,
$$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$
Does this require $X_i = 0$ a.s. or only that $EX_i=0$? Does the simple random walk not satisfy the centered equation above? The simple random walk should visit every integer infinitely many times wp1. And, if I am not mistaken, I should read $\Pr(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] ) \neq 0$ as the sum is contained in some interval infinitely often with nonzero probability. For $a=b=0$, the simple random walk (50/50 probability on 1 and -1) would therefore satisfy this. Yet I am supposed to show $X_i=0$ a.s.
Any clues to help with my misunderstanding?