Abstract
In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.
Highlights
We consider functionals that act on these clusters of relevant values and we develop useful lemmas in order to simplify the essential step to establish a Lindeberg central limit theorem (CLT) for these “cluster functionals” on stationary random fields, inspired by the definitions of Drees and Rootzén [3] and the approach of Bardet et al [4]
Under the Lindeberg condition and the convergence to zero of a sequence Tn that summarizes the dependence between the blocks of values of the random field, we prove that the finite-dimensional marginal distributions of the empirical process (4) converge to a Gaussian process
We concentrate on the first part and we introduce a measure which motivates the choice of this generalization: the iso-extremogram, which can be viewed as a correlogram for extreme values of space–time processes
Summary
Recent developments in massive data processing lead us to think differently about certain problems in statistics. These relevant values are in the cores of blocks, where the core of a block B is defined as the smaller sub-block C( B) of B which contains all relevant values of B, if they exist In this context, we consider functionals that act on these clusters of relevant values and we develop useful lemmas in order to simplify the essential step to establish a Lindeberg central limit theorem (CLT) for these “cluster functionals” on stationary random fields, inspired by the definitions of Drees and Rootzén [3] and the approach of Bardet et al [4]. Under the Lindeberg condition and the convergence to zero of a sequence Tn that summarizes the dependence between the blocks of values of the random field, we prove that the finite-dimensional marginal distributions (fidis) of the empirical process (4) converge to a Gaussian process.
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