Almost Sure Invariance Principle for Random Distance Expanding Maps with a Nonuniform Decay of Correlations

Abstract

We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

“…Another consequence of the martingale-coboundary representation is the ASIP, which in our context concerns almost sure approximation of the Birkhoff sum by Gaussians. The ASIP for random (and sequential) dynamical systems has been studied by several authors in recent years (see, for instance, [16, 20–22, 32, 50, 51]), and in this paper we will focus on the ASIP for Birkhoff sums generated by the skew product.…”

Quenched limit theorems for expanding on average cocycles

“…We emphasize that, compared with the precise class of expanding on average random dynamical systems introduced in [12], we only need the additional mild assumption that the map ω → T ω has a countable range: this is to ensure that we can apply a suitable version of the multiplicative ergodic theorem [24]. 2 We note that there are previous works devoted to limit theorems for random dynamical system that allow contracting behavior on large measure sets [1], but only under the condition that the family (T ω ) ω∈Ω only takes finitely many values (and assuming that (Ω, F , P, σ) is a Bernoulli shift),or don't require the presence of a uniform decay of correlations, such as the work of Kifer [39] (partially inspired by the work of Cogburn [14]) as well as the first author and Hafouta [21]. Roughly speaking, the main idea in those papers is to pass to the associated induced system, where the inducing is done with respect to the region of Ω on which one has the uniform decay of correlations.…”

“…The problematic aspect of this approach is the following: it is hard to ensure good control of the first entry time since a priori we cannot say anything about the region with respect to which we induce. Indeed, it remains an interesting open problem to build concrete examples to which the results of [39,21] are applicable, that don't exhibit the uniform decay of correlations.…”

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding on the average fibers

“…Probably the first papers dealing with quenched limit theorems for random dynamical systems are [36,37], and since then limit theorems for several classes of random dynamical systems were vastly studied. We refer to [3,5,15,16,17,21,22,20,18,28,30,31,51,52] for a partial list of relatively recent results of this kind. We note that in many of the examples these results are obtained for the unique measure µ such that µ ω is absolutely continuous with respect to m. However, some results hold true even for maps T ω : E ω → E σω ⊂ X which are defined on random subsets of X (see [38]), where in this case the most notable choice of µ ω is the, so called, random Gibbs measure (see [28,43]).…”

“…when T ω = T and ϕ(ω, x) = ϕ(x) do not depend on ω, many of the limit theorems follow from spectral properties of the transfer operator L T corresponding to T (namely, the dual of the Koopman operator g → g • T with respect to the underlying invariant measure µ), or from a sufficiently fast convergence of L n T towards a one dimensional projection. While several quenched limit theorems are based on an appropriate random counterpart of such spectral properties (see [15,16,17,21,22,20,18,28,30]), such annealed "spectral" techniques are not fully developed as the quenched ones. A very notable exception is the case of iid maps, discussed in the next section.…”