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Given a random walker on the number line that starts at 0 that has a 50% chance of going 1 unit in either direction every step, the walker will tend to stay on one side of the line for a while before returning to the other side.

What's the name of this phenomenon, if it has one?

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arcsine law? – Artem Jan 5 '13 at 2:40
That's for continuous functions, but pretty much. – Joe Z. Jan 5 '13 at 2:43
Yes, the best book to read about it is the first volume of Feller. – Artem Jan 5 '13 at 2:45

A discrete zero mean random walk in one dimension is recurrent; it will recross 0 infinitely often.

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if the walk is infinite. If you make an experiment looking for, say, a 1000 steps, it is more often that the person will be on one side of zero. – Artem Jan 5 '13 at 2:41
It will cross infinitely often, but I'm referring to the theorem where it tends to take very long to cross the first time. – Joe Z. Jan 5 '13 at 2:42

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