Duality and asymptotics for a class of non-neutral discrete Moran models

Thierry Huillet (1) , Martin Moehle (2)

1 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation
2 Mathematisches Institut [Tübingen]

A Markov chain X with 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 from i to i+1 with probability ( 1-i/N)* p(i/N), where 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 in [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 multi-type 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 infinity. The asymptotics of the stationary distribution of Y is studied as N tends to infinity. Examples are presented involving selection mechanisms.

Mots clés: Ancestral process; complete monotonicity; descendants; duality; Moran model; Ornstein-Uhlenbeck process

Domaines: Probabilités [math.PR]; Bio-Informatique, Biologie Systémique [q-bio.QM]; Bio-informatique [q-bio.QM]

Thierry Huillet, Martin Moehle. Duality and asymptotics for a class of non-neutral discrete Moran models. Journal of Applied Probability, 2009, 46 (3), pp.866-893. ⟨hal-00356083⟩