# Limit theorems for empirical processes of cluster functionals

Holger Drees, Holger Rootz\'en

Arxiv ID: 0910.0343•Last updated: 5/19/2020

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.

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