Volume no :7, Issue no: 2, June 2012

Journal of Statistics: Advances in Theory and Applications

DIFFUSION VERSUS JUMP PROCESSES ARISING AS SCALING LIMITS IN POPULATION GENETICS

Author's: THIERRY E. HUILLET
Pages: [85] - [154]
Received Date: March 22, 2012
Submitted by:

Abstract

When the reproduction law of a discrete branching process preserving the total size N of a population is ‘balanced’, scaling limits of the forward and backward in time processes are known to be the Wright-Fisher diffusion and the Kingman coalescent.

When the reproduction law is ‘unbalanced’, depending on extreme reproduction events occurring either occasionally or systematically, then various forward and backward jump processes, either in continuous time or in discrete time arise as scaling limits in the large N limit. This is in sharp contrast with diffusion limits, whose sample paths are continuous. We study some aspects of these limiting jump processes both forward and backward, especially the discrete-time ones. In the forward in time approach, because the absorbing boundaries are not hit in finite time, the analysis of the models together with the conclusions, which can be drawn deviate significantly from the ones available in the diffusion context.

Keywords

mutational and evolutionary processes (theory), population dynamics (theory), phylogeny (theory).