Asymptotic line-of-descent distributions
In a diffusion approximation, assume that the process is extended in the past to -\(\infty\) from a time origin 0. Let \(T_{\ell}\) be the time until there are \(\ell\) lines of descent left, beginning from n at time 0. Then the quantity \(L_ n(t)\) given by \(P(L_ n(t)\leq \ell)=P(T_{\ell}\leq t)\) represents the number of lines of descent at t in the past from the time origin. Introducing a mutation parameter \(\theta >0\) in the death-process, the author observes the random variable \(T_{\ell}\) with moment generating function \[ E(e^{\phi T_{\ell}})=\prod^{n}_{j=\ell +1}(1-2\phi /j(j+\theta -1))^{-\quad 1} \] and derives asymptotic distributions for the number of non-mutant ancestors of a sample of n from a neutral infinite alleles model: Either \(L_ n(t)\) has a normal distribution or \(n-L_ n(t)\) has a Poisson distribution as \(n\to \infty\), \(t\to 0\).
asymptotic distributions for the number of non-mutant ancestors
scientific article; zbMATH DE number 3924576