Consider a simple random walk $(X_i)_{i \in \mathbb{N}}$ on $\mathbb{Z}$ starting from the origin. Let $\tau$ be a stopping time, i.e. in order to know whether $\tau=n$ it is enough to know the first $n$ steps of the simple random walk.
Assume that the expectation of the stopping time, $E[\tau]$, is finite. Let $$ Z_n = \sum\limits_{i=0}^{n-1} \mathbb{1} \{ X_i=0 \} $$ be the number of times the random walk hits zero. Is it possible to bound from below $E[Z_\tau]$ as a function of $E[\tau]$?
As a start, the sequence $|X_n|-Z_n$ , $n=0,1,2,\ldots$, is a martingale. Here $Z_n:=\sum_{k=0}^{n-1} 1_{\{X_k=0\}}$ is the number of visits to $0$ before time $n$. Optional stopping (of this martingale and the martingale $X_n^2-n$) and Cauchy-Schwarz then lead to the estimate $\Bbb E[Z_\tau]\le\sqrt{\Bbb E[\tau]}$.