Q:

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?

A:

$(X_n^2 - n)_n$ 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.

