Duality and asymptotics for a class of nonneutral discrete Moran models

Abstract

A Markov chain X with finite state space {0,...,N} and tridiagonal transition matrix is considered, where transitions from i to i-1 occur with probability (i/N)(1-p(i/N)) and transitions from i to i+1 occur with probability (1-i/N)p(i/N). Here p:[0,1]→[0,1] is a given function. It is shown that if p is continuous with p(x)≤p(1) for all x∈[0,1] then, for each N, a dual process Y to X (with respect to a specific duality function) exists if and only if 1-p is completely monotone with p(0)=0. A probabilistic interpretation of Y in 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 on p, the process Y, properly time and space scaled, converges to an Ornstein--Uhlenbeck process as N tends to ∞. The asymptotics of the stationary distribution of Y is studied as N tends to ∞. Examples are presented involving selection mechanisms. results.