Strong invariance principles with rate for "reverse" martingales and applications

2012/09/17 by Christophe Cuny, Cuny, Christophe, Florence Merlevède +2 · 5 citations
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1209.3677

29 pages

arxiv created 2012/09/17 · arxiv updated 2012/09/18

Abstract

In this paper, we obtain almost sure invariance principles with rate of order n^1/plog^βn, 2< p≤ 4, for sums associated to a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance we treat several classes of uniformly expanding maps of the interval (for possibly unbounded functions). A general result for ϕ-dependent sequences is obtained in the course.

Cited by

• Rates of convergence in invariance principles for random walks on linear groups via martingale methods
• Limit theorems under the Maxwell-Woodroofe condition in Banach spaces
• Law of the iterated logarithm for the periodogram
• Almost sure invariance principle for sequential and non-stationary dynamical systems
• A vector valued almost sure invariance principle for time dependent non-uniformly expanding dynamical systems

Related

• Almost sure invariance principle for sequential and non-stationary dynamical systems
• Almost Sure Invariance Principles via Martingale Approximation
• Rates in almost sure invariance principle for nonuniformly hyperbolic maps
• Almost Surely Invariance Principle for Non-stationary and Random Intermittent Dynamical Systems
• Quenched Invariance Principles via Martingale Approximation