Rates of decay for the survival probability of a mutant gene

Abstract

If qk is the extinction probability of a slightly supercritical branching process with offspring distributionP kr :r = 0, 1, 2,..., then it is shown that if sup ∑r r ³ p kr , <∞, inf σ² k > 0, and m _k → 1, then 1 -q _k ~ 2(m _k -1)σ k ^-2, where m _k = ∑ r rp kr , σ² k = ∑^k r r ² p kr −m k ². This provides a simple set of sufficient conditions for the validity of a conjecture of Ewens (1969) for the survival probability of a slightly advantageous mutant gene.