Rates in almost sure invariance principle for slowly mixing dynamical systems

C Cuny (1) , J Dedecker (2) , A Korepanov (3) , Florence Merlevède (4)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 Mathematics Institute [Warwick]
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables. Our rates have form o(n γ L(n)), where L(n) is a slowly varying function and γ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed O(n 1/4). To break the O(n 1/4) barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes.

Mots clés: Strong invariance principle; KMT approximation; slowly mixing dynamical systems; Markov chain

Domaines: Probabilités [math.PR]

C Cuny, J Dedecker, A Korepanov, Florence Merlevède. Rates in almost sure invariance principle for slowly mixing dynamical systems. 2018. ⟨hal-01688705v1⟩