A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

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.

Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

“…Further research might include construction of the estimates of the rate of convergence in non-classical Lindeberg's theorem whose conditions were obtained in a recent paper [30] or some extensions to the case of dependent random summands (see [31]).…”