The Annals of Statistics

Limit theorems for empirical processes of cluster functionals

Holger Drees and Holger Rootzén

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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.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 2145-2186.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861245

Digital Object Identifier
doi:10.1214/09-AOS788

Mathematical Reviews number (MathSciNet)
MR2676886

Zentralblatt MATH identifier
1210.62051

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F17: Functional limit theorems; invariance principles 62G32: Statistics of extreme values; tail inference

Keywords
Absolute regularity block bootstrap clustering of extremes extremes local empirical processes rare events tail distribution function uniform central limit theorem

Citation

Drees, Holger; Rootzén, Holger. Limit theorems for empirical processes of cluster functionals. Ann. Statist. 38 (2010), no. 4, 2145--2186. doi:10.1214/09-AOS788. https://projecteuclid.org/euclid.aos/1278861245


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