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If $S_n/n$ converged almost surely to a finite limit, thenBut $\mathbb{P}(X_n/n>1)=\mathbb{P}(X_1>n)\sim 1/(\pi n)$. These independent-event probabilities have divergent sum, so the second Borel-Cantelli lemmas says $X_n/n>1$ infinitely often almost surely, a contradiction.

If $\kappa >8$, then for every $0<\alpha <(1-8/\kappa )/2$ and fixed $T$, there is an almost surely finite random $C$ such thatfor $|x|\le 1$ and $0<y\le 1$. The proof combines the derivative martingale, Markov inequality, a dyadic lattice, the Borel-Cantelli lemmas, and the Koebe distortion theorem.

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 $d\ge 3$.