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.
Citation
Robert J. Adler. "Weak Convergence Results for Extremal Processes Generated by Dependent Random Variables." Ann. Probab. 6 (4) 660 - 667, August, 1978. https://doi.org/10.1214/aop/1176995486
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