Limit theorems for empirical processes of cluster functionals

Abstract

Let $(X_{n,i})_{1\le i\le n,n\in\mathbb{N}}$ be a triangular array of row-wise stationary $\mathbb{R}^d$-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 $Z_n(f):=\frac{1}{\sqrt {nv_n}}\sum_{j=1}^{m_n}(f(Y_{n,j})-Ef(Y_{n,j})),$ for $v_n=P\{X_{n,i}\neq0\}$ and $f$ belonging to classes of cluster functionals, that is, functions of the blocks $Y_{n,j}$ which only depend on the cluster values and which are equal to 0 if $Y_{n,j}$ 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--\v{C}ervonenkis 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.