The Moran model with random resampling rates

In this paper we consider the two-type Moran model with N individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution ℙ on ℝ_+, and a type, either 1 or 0. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let Y^N(t) denote the empirical distribution of the resampling rates of the individuals with type 1 at time Nt. We show that if ℙ has countable support and satisfies certain tail and moment conditions, then in the limit as N→∞ the process (Y^N(t))_t ≥ 0 converges in law to the process (S(t) )_t ≥ 0, in the so-called Meyer-Zheng topology, where (S(t))_t ≥ 0 is the Fisher-Wright diffusion with diffusion constant D given by 1/D = ∫_ℝ_+ (1/r) ℙ(d r).