Simple random walk on the integer line

If the independent increments are $+1$ with probability $p$ and $-1$ with probability $q=1-p$, thenThus $\mathbb{E}{S}_n=n(p-q)$ and $\mathrm{var}(S_n)=4npq$.

For the symmetric walk and $V_{2n}=|\{0\le j\le 2n:S_j=0\}|$,Since the central binomial coefficient satisfies $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/4^k\asymp k^{-1/2}$, this expectation is of order $\sqrt{n}$.

For $d$ independent symmetric walks on $\mathbb{Z}$, the probability that all are at the origin at time $2n$ isThe Borel-Cantelli lemmas therefore show that simultaneous returns occur only finitely often almost surely when $d\ge 3$.