Modeling Neutral Evolution Using an Infinite-Allele Markov Branching Process

We consider an infinite-allele Markov branching process (IAMBP). Our main focus is the frequency spectrum of this process, that is, the proportion of alleles having a given number of copies at a specified time point. We derive the variance of the frequency spectrum, which is useful for interval estimation and hypothesis testing for process parameters. In addition, for a class of special IAMBP with birth and death offspring distribution, we show that the mean of its limiting frequency spectrum has an explicit form in terms of the hypergeometric function. We also derive an asymptotic expression for convergence rate to the limit. Simulations are used to illustrate the results for the birth and death process.