A NOTE ON THE BALANCE BETWEEN RANDOM SAMPLING AND POPULATION SIZE (ON THE 30TH ANNIVERSARY OF G. MALÉCOT'S PAPER)

Abstract

ABSTRACT Wright's model for the effects of random fluctuations in gene frequency in a population of fixed size is generalized to randomly fluctuating population size, and treated from the viewpoint of G. Malécot, using a martingale convergence theorem. The gene frequency approaches a limit, whose value depends on the actual realization, or history, of the process; that is, convergence is with probability one (or: almost surely) in statistical language. The limit does not necessarily represent a state of fixation of either allele; in particular, the limiting probability distribution is not necessarily trivial. For the special case of deterministically varying population size, a necessary and sufficient condition for such non-triviality is given.