Is the random walk squared a martingale?

Let SnS_n be a simple symmetric random walk. Is Sn2−nS_n^2 - n a martingale?

Show hints (2)+
  1. Condition one step ahead: E[(Sn±1)2∣Fn]E[(S_n\pm1)^2\mid\mathcal F_n].
  2. The +1+1 per step is exactly cancelled by subtracting nn.

Answer

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Yes - Sn2−nS_n^2 - n is a martingale

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