# GENERALISED TWO-SAMPLE U-STATISTICS AND A TWO-SPECIES REACTION-DIFFUSION MODEL

## By Mathew D. Penrose.

Consider a random variable of the form $U=\sum f(X_i,Y_j)$, where the sum is over all pairs from independent samples $(X_1,...,X_n)$ and $(Y_1,...,Y_m)$ 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.

Stochastic Processes and their Applications 55,, 57-64 (1995).