Example of a sequence of r.v.'s with constant stopping time that is not a Martingale

Question

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?

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.