Conditional gambler's ruin problem with arbitrary winning and losing probabilities with applications

Abstract

In this paper we provide formulas for the expectation of a conditional game duration in a finite state-space one-dimensional gambler's ruin problem with arbitrary winning $p(n)$ and losing $q(n)$ probabilities (i.e., they depend on the current fortune). The formulas are stated in terms of the parameters of the system. Beyer and Waterman [Mathematics Magazine, 50(1), 1977] showed that for the classical gambler's ruin problem the distribution of a conditional absorption time is symmetric in $p$ and $q$. Our formulas imply that for non-constant winning/losing probabilities the expectation of a conditional game duration is symmetric in these probabilities (i.e., it is the same if we exchange $p(n)$ with $q(n)$) as long as a ratio $q(n)/p(n)$ is constant. Most of the formulas are applied to a non-symmetric random walk on a circle/polygon. Moreover, for a symmetric random walk on a circle we construct an optimal strong stationary dual chain -- which turns out to be an absorbing, non-symmetric, birth and death chain. We apply our results and provide a formula for its expected absorption time, which is a fastest strong stationary time for the aforementioned symmetric random walk on a circle. This way we improve upon a result of Diaconis and Fill [The Annals of Probability, 18(4), 1990], where strong stationary time -- however not the fastest -- was constructed. Expectations of the fastest strong stationary time and the one constructed by Diaconis and Fill differ by 3/4, independently of a circle's size.