Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

Abstract

In this paper, we obtain almost sure invariance principles with rate of order n 1/p log β n, 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. (2010): 37E05, 37C30, 60F15. Mathematics Subject Classification

Almost sure invariance principle for random piecewise expanding maps

“…In situations when this method is applicable, it was pointed out in [11] that it gives better error rates in ASIP when compared to those obtained in [17,18]. Finally, we mention the recent important papers by Cuny and Merlevede [4], Korepanov, Kosloff and Melbourne [16], Korepanov [15] as well as Cuny, Dedecker Korepanov and Merlevede [2,3] in which the authors further improved the error rates in ASIP for a wide class of (nonuniformly) hyperbolic deterministic dynamical systems.…”

Almost surely invariance principle for non-stationary and random intermittent dynamical systems

“…Surprisingly, this condition can be verifed by our system considered here. Besides, due to nonuniformity of our system, the error rate of our ASIP is just slightly less than 1 2 (not 1 4 in [2]). So we will not give an explicit formula for it.…”

“…In this paper, the same system as [6] is considered and some of its properties are improved, namely the stronger statistical property (ASIP) is obtained. Our construction for Gaussian variable in ASIP is close to Proposition 2.1 in [2], that is, applying Skorohod embedding to tail series, but we won't impose strong conditions like (2.1), (2.2) in [2], which loses lots of information of Skorohod embedding. Instead, we will give a sharp condition for ASIP (see our Lemma 4.4).…”

“…Same approach is applied to random dynamical system with sufficient hyperbolicity in [3]. Based on Skorokhod embedding technique, Cuny and Merlevède [2] recently showed that reverse martingale differences satisfy ASIP under some conditions. The error is shown to be essentially bounded due to the presence of a spectral gap in the transfer operator on a Banach space continuously injected in L ∞ .…”

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

“…There has been a great deal of work on the ASIP in the probability theroy, see for instance [44,3,29,51,50,22,55,23]. In the context of the stationary process generated by bounded Hölder observables over smooth dynamical systems with singularities, the ASIP was first shown by Chernov [12] for Sinai dispersing billiards.…”