Asymptotic Tail Behavior of Uniform Multivariate Empirical Processes

Let $\alpha_n$ be the empirical process of independent uniformly distributed random vectors on the unit square $I^2$. We study the asymptotic distribution of the random variable $\sup|\alpha_n(s,t)|/(s^\nu t^\mu L(s)G(s))$ when $\sup$ is taken over various subintervals of $I^2$. We show that in the case of $-\infty < \mu, \nu < 1/2$ the limit is given in terms of a two-time parameter Wiener process, and for $1/2 < \mu, \nu < \infty$ it is determined by a Poisson process.