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?
what is the behaviour of moving dot with 50% chance to go left or right?
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$\begingroup$
probability
random-walk
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0is there a reason why someone gave -1 to this question? – 2012-07-26
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0It is a symmetric random walk and it will, with positive probability, return to any state infinitely often. – 2012-07-26
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0If 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) – 2012-07-26
1 Answers
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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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2On the other hand, it has probability zero of staying bounded. – 2012-07-26
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1@Gerry: A fact which is included in the statement that the random walk *visits every point infinitely often with probability one*. – 2012-07-26
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0However the waiting time for it to return to any given state has infinite mean. – 2012-07-26