Almost Sure Invariance Principle for Random Piecewise Expanding Maps

The objective of this note is to prove the almost sure invariance principle (ASIP) for a large class of random dynamical systems. The random dynamics is driven by an invertiblemeasure preserving transformation σ of (Ω,F ,P) called the base transformation. Trajectories in the phase space X are formed by concatenations f ω := fσn−1ω ◦ · · · ◦ fσω ◦ fω of maps from a family of maps fω : X → X, ω ∈ Ω. For a systematic treatment of these systems we refer to [2]. For sufficiently regular bounded observables ψω : X → R, ω ∈ Ω, an almost sure invariance principle guarantees that the random variables ψσnω ◦ f ω can be matched with trajectories of a Brownian motion, with the error negligible compared to the length of the trajectory. In the present paper, we consider observables defined on some measure space (X,m) which is endowed with a notion of variation. In particular, we consider examples where the observables are functions of bounded variation or quasiHölder functions on a compact subset X of R. We emphasize that our setting is quite similar to that in [3], where the maps fω are called random Lasota-Yorke maps. In a more general setting and under suitable assumptions, Kifer proved in [11] central limit theorems (CLT) and laws of iterated logarithm; we will briefly compare Kifer’s assumptions with ours in Remark 2 below. In [11, Remark 2.7], Kifer claimed without proof (see [11, Remark 4.1]) a random functional CLT, i.e. the weak invariance principle (WIP), and also a strong version of the WIP with almost sure convergence, namely the almost sure invariance principle (ASIP), referring to techniques of Philip and Stout [13]. Here we present a proof of the ASIP for our class of random transformations, following a method recently proposed by Cuny and Merlèvede [6]. This method is particularly powerful when applied to non-stationary dynamical systems; it was successfully used in [9] for a large class of sequential systems with some expanding features and for which only the CLT was previously known [5]. We stress that ω-fibered random dynamical systems discussed above are also non-stationary since we use ω-dependent sample measures (see below) on the underlying probability space.