Conditions for fixation of an allele in the density-dependent Wright-Fisher models

A density-dependent Wright-Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizes Mn vary without density dependence then the condition ΣMnab21’-¹=• is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.