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Suppose $X_1,X_2,\ldots$ are i.i.d. $\mathbb Z$-valued random variables such that the random walk $$S_n=\sum_{i=1}^nX_i$$ is recurrent with some period $k\geq1$ (i.e., $\Pr[S_n=0]>0$ if and only if $n\in k\mathbb N$). Consider the first return time to zero $$T=\inf\{n\geq1:S_n=0\}.$$

Question. Is it always the case that the probabilities $p_n:=\Pr[T=kn]$, are eventually nonincreasing in $n$? (That is, there is some $N$ large enough so that $p_n\geq p_{n+m}$ for all $m$ and $n\geq N$.)

In two simple cases, i.e., if $X_i$ are uniform on $\{-1,1\}$ or $\{-1,0,1\}$, then we can compute the $p_n$ exactly by using Catalan or Motzkin numbers, and we see in those cases that the $p_n$ are strictly decreasing.

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