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For a 1 D random walk on $Z$ axis, starting at $z=0$, equal probability to go to right or left, what is the probability that during the first k steps the walker's position remains $z\leq m$?

This is also called the survival probability. I can write is as a sum of first passage time probabilities, but I am looking for a closed form.

Thanks,

Albert
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  • I meant during the first k steps the position should be at m or lower. (I edited the question). Thanks for your comment. – Albert Aug 19 '15 at 21:52
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    You can always use the reflection principle to get an explicit expression as in my answer here: http://math.stackexchange.com/questions/4234/random-walk-0/4235#4235 I don't think you will get a simpler closed form than this. –  Aug 19 '15 at 23:50
  • @ByronSchmuland: Wow, a question and answer with a four-digit number from $2010$ -- early math.SE history :-) Actually this is a duplicate of that question, no? – joriki Aug 20 '15 at 04:37
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    @joriki It was my first answer on MSE. Ancient history! –  Aug 20 '15 at 12:34
  • Byron, Thank for that. I saw your answer there. You mean we don't have a closed for for the CDF of the occupancy probability? $P(0\leq Xt<|b|)$ – Albert Aug 21 '15 at 01:48
  • @ByronSchmuland hank for that. I saw your answer there. You mean we don't have a closed for for the CDF of the occupancy probability? P(0≤Xt<|b|) – Albert Aug 21 '15 at 17:56

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