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If the universe goes on long enough..........

Posted By: Maik Stiebler
Date: Tuesday, 18 May 2010, at 8:01 p.m.

In Response To: If the universe goes on long enough.......... (Steve Mellen)

but I've seen it proven conclusively that a random walk always reaches one of the boundaries given an infinite duration

I find that relatively unsurprising, given that the standard deviation of the distance from the origin that the random walk (assuming unit steps and zero mean) reaches after N steps is sqrt(N), which obviously can get very much larger than the distance of the boundaries from the origin. A more dazzling fact is that in an infinitely long random walk, the absolute value of the distance reached at step N divided by sqrt(N), i.e. the number of sd's, will also reach any given boundary with probability 1. That's a consequence of the "Law of the iterated logarithm". For backgammon players it means that even a rollout stopping criterion of, say, 1000 jsd's does not guarantee any significance. Admittedly that's only true if your willing to do a really huge lot of trials.

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