How many times does a line walk revisit zero?

Question

A token does a symmetric \pm 1 walk on the number line, starting at 0 and continuing forever. On average, how many times does the token return to 0 over its infinite journey?

Accepted Answer

The 1D walk is recurrent, so each time it sits at 0 it is certain to come back again. With every return guaranteed, the returns never stop: the token visits 0 again and again without bound, so the expected number of returns is infinite. E[	ext{number of returns}] = \infty. You can also see it from the return chances: the expected number of visits to 0 equals the sum over all times n of the probability of being at 0 at step n, and that sum diverges in 1D (and in 2D). In 3D, by contrast, the walk is transient and the same sum converges, so there the expected number of returns is finite. Takeaw…

How many times does a line walk revisit zero?

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