= Oscillation of a bounded centered iid random walk
{title2=$\liminf S_n=-\infty,\quad\limsup S_n=+\infty$}
Let $S_n=\sum_{j=1}^nX_j$, where the increments are nonconstant <independent and identically distributed random variables> with zero <expected value> and $|X_j|\leq c$. Then the <random walk> oscillates between arbitrarily large positive and negative levels <almost surely>. For $T_a=\inf\{n:S_n\geq a\}$, the <stopped martingale> $a+c-S_{n\wedge T_a}$ is nonnegative. The <martingale convergence theorem> implies that $S_n$ converges finitely on $\{T_a=\infty\}$. However, some $\delta>0$ has $\mathbb P(|X_1|>\delta)>0$, and the <Borel-Cantelli lemmas> imply $|X_n|>\delta$ infinitely often. Hence finite convergence is impossible. Apply this to every positive integer $a$ and to $-S_n$.
The <bounded increments> also imply infinitely many visits to $[-c/2,c/2]$: a path cannot cross from above $c/2$ to below $-c/2$ without entering this interval. Identical distribution cannot simply be dropped. For independent symmetric signs $\varepsilon_j$, the increments $X_j=c4^{-(j-1)}\varepsilon_j$ satisfy $|S_n|\geq2c/3$ for every $n$, even though they are nonzero, independent, symmetric and uniformly bounded.
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