# Stopping time and martingale for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a stopping time, while $\tau-1$ may not be one. My question is: Can I say $E[S_{\tau-1}X_{\tau}]=0$? Further, $E[S_{\tau-1}^2]=E[\tau-1]$? Thanks for any hints, discussion and help.

• It seems very easy. But I have not found related results. Wish some hints. – Richard Aug 21 '13 at 0:43