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Limit Theorems for Empirical Processes of Cluster Functionals

Journal article
Authors Holger Drees
Holger Rootzén
Published in Annals of statistics
Volume 38
Issue 4
Pages 2145-2186
ISSN 0090-5364
Publication year 2010
Published at Department of Mathematical Sciences, Mathematical Statistics
Pages 2145-2186
Language en
Keywords absolute regularity, block bootstrap, clustering of extremes, extremes, local empirical processes, rare events, tail distribution function, uniform central limit theorem
Subject categories Applied mathematics


Let (X-n, i) 1 <= i <= n,m is an element of N be a triangular array of row-wise stationary R-d-valued random variables. We use a "blocks method" to define clusters of extreme values: the rows of (X-n, i) are divided into m(n) blocks (Y-n, j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes Z(n)(f) := 1/root nv(n) Sigma(mn)(j=1) (f(Y-n,Y- j) - Ef(Y-n,Y- j)), for v(n) = P{X-n,X- i not equal 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Y-n,Y- j which only depend on the cluster values and which are equal to 0 if Y-n,Y- j does not contain a cluster. Conditions for finite-dimensional convergence include beta-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either "bracketing entropy" or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik-Cervonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

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