Rates in almost sure invariance principle for slowly mixing dynamical systems

Abstract

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-towerlike Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes. Theorem 1.3. Let γ ∈ (0, 1/2) and ϕ : [0, 1] → R be a Hölder continuous observable with ϕ dµ = 0. For the map (1.1), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ (log n) γ+ε ) for all ε > 0. For the map (1.5), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ ).The rates in Theorem 1.3 are optimal in the following sense:Proposition 1.4. Let f be the map (1.1). There exists a Hölder continuous observable ϕ with ϕ dµ = 0 such that lim sup n→∞ (n log n) −γ |S n (ϕ) − W n | > 0 for all Brownian motions (W t ) t≥0 defined on the same (possibly enlarged) probability space as (S n (ϕ)) n≥0 . Hence, one cannot take ε = 0 in Theorem 1.3.Remark 1.5. If c 2 = 0, the rate in the ASIP can be improved to O(1). Indeed, then it is well-known that ϕ is a coboundary in the sense that ϕ = u−u•f with some u : [0, 1] → R. By [7, Prop. 1.4.2], u is bounded, thus S n (ϕ) is bounded uniformly in n.Remark 1.6. It is possible to relax the assumption that ϕ is Hölder continuous. As a simple example, Theorem 1.3 holds if ϕ is Hölder on (0, 1/2) and on (1/2, 1), with a discontinuity at 1/2. See Subsection 4.3 for further extensions.Remark 1.7. Intermittent maps are prototypical examples of nonuniformly expanding dynamical systems, to which our results apply in a general setup, and so does the discussion of rates preceding Theorem 1.3. We focus on the maps (1.1) and (1.5) for simplicity only, and discuss the generalization in Section 5.

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.…”

Rates in almost sure invariance principle for quickly mixing dynamical systems

“…The method of [14] only works for exponential decay of return times. It was improved by the authors of this paper in [7], where we obtained a significantly more general result which covers maps with polynomial decay of return times, such as intermittent maps: 7]). Let f : X → X be a nonuniformly expanding map (see Section 2.1) with the reference measure m, return time τ and physical invariant measure µ.…”

“…In Section 3, we prove Theorem 1.6. The proof is based on the construction of approximating Brownian motion from [3] as in [8] and [7]. One of the crucial tools is the ASIP for processes with independent (but not necessarily identically distributed) increments by Sakhanenko [25, Thm.…”

“…Following [7,14], for nonuniformly expanding dynamical systems, the random process S n (ϕ) can be redefined on a Markov shift without changing its distribution. The structure of the Markov shift is as follows.…”

“…Then d is a separation metric on Ω, the separation time counted in terms of returns to S 0 . We proved [7,14] that given a nonuniformly expanding dynamical system f : X → X with a Hölder continuous observable ϕ : X → R, there exists a Markov chain as above and a Hölder continuous function ψ : Ω → R, such that…”