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

Jérôme Dedecker (1) , Herold G. Dehling (2) , Murad Taqqu (3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 RUB - Ruhr University Bochum = Ruhr-Universität Bochum
3 Department of Mathematics and Statistics [Boston]

We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square inegrable 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.

Mots clés: Long-range dependence; intermittency; empirical process

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Jérôme Dedecker, Herold G. Dehling, Murad Taqqu. Weak convergence of the empirical process of intermittent maps in L2 under long-range dependence. Stochastics and Dynamics, 2015, 15 (2), 29pp. ⟨hal-00903837⟩