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


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.


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Robert J. Adler. "Weak Convergence Results for Extremal Processes Generated by Dependent Random Variables." Ann. Probab. 6 (4) 660 - 667, August, 1978.


Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0377.60027
MathSciNet: MR494408
Digital Object Identifier: 10.1214/aop/1176995486

Primary: 60F05
Secondary: 60B10 , 60G10 , 60G15

Keywords: $K$-dimensional extremal process , Dependent stationary sequence , Gaussian sequence , Skorohod topology , two-dimensional Poisson process , vague topology , weak convergence

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • August, 1978
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