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