Rates of decay for the survival probability of a mutant gene

If qk is the extinction probability of a slightly supercritical branching process with offspring distribution P_kr : r = 0, 1, 2,..., then it is shown that if sup _rr³p_kr, < , inf ²_k > 0, and m_k 1, then 1 – q_k 2(m_k–1)_k^–2, where m_k= _rrp_kr, ²_k = ^k_rr²p_kr– m_k². 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.Research supported in part by NSF grants DMS-8803639 and DMS-9007182