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Consider a simple symmetric random walk on $\mathbb{Z}$ starting from the origin. Let $\tau$ be a stopping time for the simple random walk and let $Z_n$ be the number of returns to zero of the simple random walk from time $0$ to time $n$.

Is it true that if $E[\tau]$ is infinite, then $E[Z_\tau]$ is also infinite? The intuition should be that, if infinite time is weighted, then the walk will visit the origin infinite number of times.

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No. The time $\tau$ of first return to the origin is a stopping time with $\mathsf E[\tau]=\infty$, but $\mathsf E[Z_\tau]=1$.

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