Empirical central limit theorems for ergodic automorphisms of the torus

2012/10/12 by J. Dedecker, Dedecker, J., F. Merlevède +3
Mathematics · #37D30 #60F17 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:37D30 #msc:60F17

paper · pdf · doi:10.48550/arxiv.1210.3546

32 pages

arxiv created 2013/09/27 · arxiv updated 2013/10/01

Abstract

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o Tⁱ)_{i ≥ 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.

Related

• Rates of convergence in the strong invariance principle for non adapted sequences. Application to ergodic automorphisms of the torus
• Rates in the strong invariance principle for ergodic automorphisms of the torus
• Decay of correlation for expanding toral endomorphisms
• Empirical Processes of Multidimensional Systems with Multiple Mixing Properties
• Convergence theorems for barycentric maps