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I am looking for a closed form equation for the expected number of times symmetric binary walk hits zero value in K steps (at the limit of large number of steps).

In other words:

$x[0]=0$

$x[k]=x[k-1]+b[k]$,

where $b[k]$ is $-1$ or $1$ (probability $0.5$ each).

What is the expected value of the number of times $x[i]==0$ (where i=0 to K), ie number of zero-value hitting in the first K steps.

(A Reference to a textbook or paper would be nice)

Thanks!

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    Textbook and paper references are *very* nice, I agree. If you are a university student, this sounds exactly like the sort of thing your library of mathematical literature would contain. – 2017-01-10
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    The closed form equation that you want is in the answer here: http://math.stackexchange.com/questions/981242/expected-number-of-returns-to-zero-in-a-symmetric-random-walk-closed-form – 2017-01-11

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