Weak Convergence Results for Extremal Processes Generated by Dependent Random Variables

Abstract

In this paper we consider a stationary sequence $\{X_n, n \geqq 1\}$ satisfying weak dependence restrictions similar to those recently introduced by Leadbetter. Suppose $a_n$ and $b_n > 0$ are norming constants for which $\max\{X_{n1},\cdots, X_{nn}\}$ converges in distribution, where $X_{nk} = (X_k - b_n)/a_n$. Define a sequence of planar processes $I_n(B) = \sharp\{j: (j/n, X_{nj}) \in B, j = 1,2,\cdots, n\}$, where $B$ is a Borel subset of $(0, \infty) \times (-\infty, \infty)$. Then the $I_n$ converge weakly to a nonhomogeneous two-dimensional Poisson process possessing the same distribution as for independent $X_j$. Applying the continuous mapping theorem to this result generates a variety of further results, including, for example, weak convergence of the order statistics of the $X_n$ sequence. The dependence conditions are weak enough to include the Gaussian sequences considered by Berman.