Research interview, usually as a lead-in to discussing stationarity.
Returning to the origin
The question
A symmetric random walk on the integers starts at zero. Does it return to zero with probability one, and what is the expected time to return?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
It returns with probability one, but the expected return time is infinite.
The probability of being at zero after 2n steps is the central binomial coefficient over 4^n, which behaves like 1/sqrt(pi n).
The sum of those probabilities diverges, so by the recurrence criterion the walk is recurrent: it returns with probability one.
The return time distribution has a tail behaving like n^(-3/2), and the sum of n times that tail diverges, so the mean return time is infinite.
This is called null recurrence, and it is the cleanest example of a thing that certainly happens yet has no finite expected waiting time. In three dimensions the walk is not even recurrent.
What it is testing
Whether recurrence and finite expectation are separate ideas in your head.
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Preparing properly
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