An Asymptotic Result in a Pure Birth Stochastic Process

Abstract

The F 1 2 hypergeometric function appears in a wide variety of applications in probability and statistics. This article is motivated from an important application that arises in stochastic processes. A birth and death process that starts with a single species and birth rate larger than the death rate may lead to trees of species or genera. Under standard assumptions on the process, the probability of the size of a genera is a discrete distribution that can easily be computed at time t from the origin. However, when the process leads to several sub-lineages of trees that originated at different times from the origin, then the distribution of the size of the population commonly referred to as taxon size distribution of the genus size must be weighted by the different times that the genera originate. It was shown in [1], that under the exponential distribution for time t the taxon distribution may be expressed as an F 1 2 (, ; + ;) hypergeometric function. This form has not been exploited in the literature. In this article we consider an asymptotic expansion of this 2 F 1 for large x in terms of generalized Bernoulli polynomials that can be computed up to any degree of accuracy.