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

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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).