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August 2010 Limit theorems for empirical processes of cluster functionals
Holger Drees, Holger Rootzén
Ann. Statist. 38(4): 2145-2186 (August 2010). DOI: 10.1214/09-AOS788

Abstract

Let (Xn,i)1≤in,n∈ℕ be a triangular array of row-wise stationary ℝd-valued random variables. We use a “blocks method” to define clusters of extreme values: the rows of (Xn,i) are divided into mn blocks (Yn,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 $Z_{n}(f):=\frac{1}{\sqrt{nv_{n}}}\sum_{j=1}^{m_{n}}(f(Y_{n,j})-Ef(Y_{n,j}))$, for vn = P{Xn,i ≠ 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Yn,j which only depend on the cluster values and which are equal to 0 if Yn,j does not contain a cluster. Conditions for finite-dimensional convergence include β-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–Červonenkis 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.

Citation

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Holger Drees. Holger Rootzén. "Limit theorems for empirical processes of cluster functionals." Ann. Statist. 38 (4) 2145 - 2186, August 2010. https://doi.org/10.1214/09-AOS788

Information

Published: August 2010
First available in Project Euclid: 11 July 2010

zbMATH: 1210.62051
MathSciNet: MR2676886
Digital Object Identifier: 10.1214/09-AOS788

Subjects:
Primary: 60G70
Secondary: 60F17 , 62G32

Keywords: absolute regularity , block bootstrap , clustering of extremes , Extremes , local empirical processes , Rare events , tail distribution function , uniform central limit theorem

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 4 • August 2010
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