Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models

Abstract

A Markov chainXwith finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions fromitoi-1 occur with probability (i/N)(1-p(i/N)) and transitions fromitoi+1 occur with probability (1-i/N)p(i/N). Herep:[0,1]→[0,1] is a given function. It is shown that ifpis continuous withp(x)≤p(1) for allx∈[0,1] then, for eachN, a dual processYtoX(with respect to a specific duality function) exists if and only if 1-pis completely monotone withp(0)=0. A probabilistic interpretation ofYin terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions onp, the processY, properly time and space scaled, converges to an Ornstein–Uhlenbeck process asNtends to ∞. The asymptotics of the stationary distribution ofYis studied asNtends to ∞. Examples are presented involving selection mechanisms. results.