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

Author

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  • José G. Gómez-García

    (UNICAEN, CNRS, LMNO, Laboratory of Mathematics Nicolas Oresme, UFR des Sciences, Bld Maréchal Juin, Campus 2, Normandie Université, 14032 Caen, France
    The first author was funded by the Normandy Region RIN program.)

  • Christophe Chesneau

    (UNICAEN, CNRS, LMNO, Laboratory of Mathematics Nicolas Oresme, UFR des Sciences, Bld Maréchal Juin, Campus 2, Normandie Université, 14032 Caen, France)

Abstract

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes ( Z n ( 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 T n 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.

Suggested Citation

  • José G. Gómez-García & Christophe Chesneau, 2021. "A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields," Mathematics, MDPI, vol. 9(3), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:212-:d:484224
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    References listed on IDEAS

    as
    1. Davis, Richard A. & Mikosch, Thomas, 2008. "Extreme value theory for space-time processes with heavy-tailed distributions," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 560-584, April.
    2. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    3. Segers, J.J.J., 2003. "Functionals of Clusters of Extremes," Other publications TiSEM 948d700b-a923-4068-b4ad-3, Tilburg University, School of Economics and Management.
    4. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    5. Segers, J.J.J., 2003. "Functionals of Clusters of Extremes," Discussion Paper 2003-48, Tilburg University, Center for Economic Research.
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    Cited by:

    1. Ruslan Gabdullin & Vladimir Makarenko & Irina Shevtsova, 2021. "Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem," Mathematics, MDPI, vol. 9(5), pages 1-32, March.

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