Hypergeometric solution to a gambler's ruin problem with a nonzero halting probability

Abstract

This paper treats of a kind of a gambler's ruin problem, which seeks the probability that a random walker first hits the origin at a certain time. In addition to a usual random walk which hops either rightwards or leftwards, the present paper introduces the `halt' that the walker does not hop with a certain probability. The solution to the problem can be obtained exactly using a Gauss hypergeometric function. The moment generating function of the duration is also calculated, and a calculation technique of the moments is developed. The author derives the long-time behavior of the ruin probability, which exhibits power-law behavior if the walker hops to the right and left with equal probability.