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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.