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If a dot is moving (from zero) left or right, by one, with 50% chance to go left or right - is it going to go to the +inf or -inf when it has infinite moves?

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is there a reason why someone gave -1 to this question? – Nole Taoci Jul 26 '12 at 8:15
It is a symmetric random walk and it will, with positive probability, return to any state infinitely often. – Stefan Hansen Jul 26 '12 at 8:25
If you are sure, why not post it as an answer? I am asking this because I remembered from school that we had a proof saying yes to this question, when learning probability, but it doesn't make to much sense when you compare moving left/right to the (for example) flipping coins - we expect it will be around zero (equal number of heads/tails) – Nole Taoci Jul 26 '12 at 8:34

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This is the simple symmetric random walk on the integers. It is well known that this walk is recurrent, and so visits every point infinitely often with probability one. In particular, it has probability zero of converging to either $+\infty$ or $-\infty$. Proofs can be found here, or in most introductory textbooks on random processes.

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On the other hand, it has probability zero of staying bounded. – Gerry Myerson Jul 26 '12 at 12:05
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@Gerry: A fact which is included in the statement that the random walk visits every point infinitely often with probability one. – Did Jul 26 '12 at 12:21
However the waiting time for it to return to any given state has infinite mean. – Stefan Hansen Jul 26 '12 at 14:02

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