Square-minus-time martingale of a simple symmetric random walk ($S_n^2-n$)

For a simple symmetric random walk, $S_n^2-n$ is a martingale because the next increment has conditional mean zero and conditional square one.

Let $S_n$ be a simple symmetric random walk started at a deterministic value $S_0=s$, and let $T$ be a nonnegative integer-valued stopping time with $\mathbb{E}T<\infty$. The stopped second-moment identity isIn particular, $\mathbb{E}{S}_T^2=\mathbb{E}T$ requires the zero start $s=0$.

For a direct argument, write $S_{T\wedge n}-s={\sum }_{k=1}^n{\xi }_k{1}_{\{T\ge k\}}$. The indicators depend only on the past. Centered independent increments therefore give zero cross terms andFor $m>n$, the same calculation gives the squared L2 norm of the difference as $\mathbb{E}[(T\wedge m)-(T\wedge n)]$, which tends to zero. Thus the stopped positions converge in L2 space, as well as eventually pointwise to $S_T$, and both their first and second moments pass to the limit. This supplies the required justification for an unbounded stopping time; almost-sure convergence alone would not pass the second moment.