Generalised two-sample U-statistics and a two-species reaction-diffusion model

Consider a random variable of the form U = ∑ f(Xi, Yj), where the sum is over all pairs from independent samples (X1, …, Xn) and (Y1, …, Ym) from two (possibly different) distributions, and f is a given function which may depend on n. We discuss possible limits for the distribution of U when n becomes large with n(m+n) approaching a fixed limit. We discuss an application to a Brownian motion for the irreversible two-species, diffusion-controlled chemical reaction.