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Weak convergence of the empirical process of intermittent maps in 𝕃² under long-range dependence

Jérôme Dedecker

Université Paris Descartes, Sorbonne Paris Cité, Laboratoire MAP5 and CNRS UMR 8145, 75016 Paris, France

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Herold Dehling

Faculty of Mathematics, Ruhr-University Bochum, 44780 Bochum, Germany

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, and

Murad S. Taqqu

Department of Mathematics, Boston University, Boston, MA 02215, USA

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https://doi.org/10.1142/S0219493715500082Cited by:8 (Source: Crossref)

Abstract

We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square integrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér–von-Mises statistic.

Keywords:

AMSC: 60F17, 60E07, 37E05

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