Assume is a set with , such that diverges. Let , each with probability . What can we say about when goes to infinity? Must there, for example, be infinitely many sign changes with probability ? It can’t be hard to show this for bounded , but what about for example? I honestly have no clue whatsoever.

EDIT 22 Februari 2011

I can show that there is a positive probability that for any , for large enough; Let be the smallest integer such that (note that this exists since the sum of is assumed to go to infinity).

Then either the probability that is at least or the probability that is at least (*). Assume the first case (second case goes analogous), then we have:

(*) It is fun to note that both are the case, independent of . This can be used to show that for all

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