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Let $R$ be the number of returns to $v$ after time $0$. Since $v$ is a recurrent state, the probability of another return after each visit is one. By the stated geometric distribution description, ${\mathbb{P}}_v(R\ge k)=1$ for every positive integer $k$. Thus $R=\infty$ almost surely and ${\mathbb{E}}_{v}R=\infty$.

On the other hand, writing each visit as an indicator random variable and applying the monotone convergence theorem to the partial sums givesThe left side is infinite, proving the recurrence criterion by return probabilities in the required direction:

For the simple random walk on the integer line, a return to the origin is possible only at an even time. At time $2n$, exactly $n$ of the increments must be $+1$, so the return probability isThe supplied factorial bounds, equivalently the order estimate in the Stirling formula, giveConsequently ${\sum }_n{p}_{2n}$ diverges. By the recurrence criterion by return probabilities, the origin is a recurrent state; translation invariance then makes the whole walk recurrent.

For three independent walks, the probability that all three are at the origin at time $2n$ isThis P-series is convergent, so the first of the Borel-Cantelli lemmas says that simultaneous returns occur only finitely often with probability one. Therefore the requested probability is