Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables

Abstract

For an array $\{X_{ni}\}$ of independent, uniformly null random variables, several necessary and sufficient conditions are given for the convergence in distribution of its extremal process $\mathbf{M}_n = (M^1_n, M^2_n, \cdots)$ as $n \rightarrow \infty$, where $M^k_n(t) = k$th largest $\{X_{ni}: i/n \leq t\}, t > 0$. It is shown that if $\mathbf{M}_n$ converges, then its limit is an extremal process of a Poisson process on the plane. The limit cannot be an extremal process of a non-Poisson, infinitely divisible point process, which is possible for certain stationary variables. A characterization of the convergence of $\mathbf{M}_n$, without the uniformly null assumption, is also given.