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.

If $T$ is a stopping time with finite mean, thenStopping first at $T\wedge n$ and using the $L^2$ martingale convergence theorem justifies passage to an unbounded $T$.