Could anybody give me a simple example of a sequence of random variables $(X_{n})_{n \geq 0}$ that has constant expectation, but is not a martingale?

  • $\begingroup$ Question amended. $\endgroup$
    – user100463
    Dec 14, 2013 at 19:42

1 Answer 1


If $(S_n)$ is a simple symmetric random walk on the integers, then $X_n:=S_n^3$ is a mean zero process that is not a martingale.

  • $\begingroup$ +1. This might well be one of the simplest examples--hence one of the best. $\endgroup$
    – Did
    Dec 14, 2013 at 18:36
  • $\begingroup$ @Did Thanks for the kind words (again!) $\endgroup$
    – user940
    Dec 14, 2013 at 18:38

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