# Is there a version of the arcsine rule for biased coins?

I'm just reading about the arcsine rule for the first time in the context of coin flipping, i.e., that one player is very likely to lead a large percentage of the time in a finite number of trials.

Question: Is whether there is a generalized version of this for biased coin?

I tried finding something via a search engine, but a lot of the literature is in the context of academic papers on Brownian motion, so I'm unable to make heads or tails of it (pun unintended).

Thanks!

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Can you be a little more specific regarding which arcsine law you are interested in? What you currently describe almost sounds more like the ballot theorem. There are generalizations to the arcsine law regarding the proportion of intervals between ties in which one player is leading. Note that in an biased coin setting there is a nonnegligible probability of never returning to zero. –  cardinal Apr 13 '12 at 19:44
I'm only now attempting to learn prob/stat so I'm going to pre-apologize if this isn't clear. I was reading that in a finite coin flipping/random walk setting that you're very likely to be on one side of 0 for a majority of the steps. And that this probability is given by a formula involving arcsin. My question was really along the lines of: assume you have a 1/3-2/3 coin, can you state anything about the probability of being on one side of the "new zero"="1/3rd". I hope that's reasonably coherent. –  Goren Apr 13 '12 at 19:52
Hi Goren. Thanks. Yes, that's helpful in determining what you are interested in and no need to apologize at all! :) –  cardinal Apr 13 '12 at 20:26