# Expected distance of the random walk at a random stopping time

Let $X(i)$ be a simple symmetric random walk in $\mathbb{Z}$ starting from the origin. Let $\tau$ be a stopping time for the random walk, namely for any $k \in \mathbb{N}$, the event $\{\tau=k\}$ depends only on the first $k$ steps of the random walk. An example of $\tau$ might be the first time the simple random walk hits a given vertex $m \in \mathbb{Z}$.

Assume we know the expectation $E[\tau]$ and that $E[ \, X(\tau) \, ] = 0$. Is there a way to infer some information on $E[ \, | X(\tau) | \, ]$, the expected distance of the random walk from the origin at time $\tau$? In other words, are $E[\tau]$ and $E[ \, \, | X(\tau) | \, \, ]$ connected by some inequality?

## 1 Answer

$(X_n^2 - n)_n$ is a martingale. By Jensen's inequality, $\lvert E[X_\tau]\rvert \leq \left(E[\lvert X_\tau\rvert^2]\right)^{\frac{1}{2}}$. If you can justify $E[X_{\tau}^2] = E[\tau]$, you would have $\lvert E[X_\tau]\rvert \leq \left(E[\tau]\right)^{\frac{1}{2}}$. It is possible that you get a stronger connection through some other considerations. Maybe look at Wald's identity.

• Thanks for the comment. I added an additional condition that I know, namely that $E[X(\tau)]=0$. Mar 4, 2018 at 14:02
• It is hard to say anything new based on the additional condition that $E[X_{\tau}] = 0$. Think of the following stopping time $\tau = \min\{n: X_n = b \text{ or } X_n = -b\}$ for some $b > 0$. Then $E[\tau] = b^2$, $E[X_{\tau}] = 0$ and $E[\lvert X_{\tau}\rvert] = b$. This satisfies the inequality $E[\lvert X_{\tau}\rvert] \leq \left(E[X_{\tau}]\right)^{\frac{1}{2}}$ with an equality. It is tempting to think that such an equality would hold for any stopping time with $E[X_\tau] =0$ but I doubt this is true. Maybe if $\tau$ is a stopping time wrt the natural filtration of $X$. Mar 4, 2018 at 14:25