LIMIT THEOREMS FOR EMPIRICAL PROCESSES OF CLUSTER FUNCTIONALS

• 成果类型: Article
• 署名作者: Drees, Holger; Rootzen, Holger
• 署名单位: University of Hamburg; Chalmers University of Technology; University of Gothenburg
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS788
• 发表日期: 2010
• 页码: 2145-2186
• 关键词: tail; CONVERGENCE; uniform

摘要

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.