On the Asymptotic moments and Edgeworth expansions for some processes in random dynamical environment

TL;DR

This paper establishes the existence and computation of asymptotic moments for certain random dynamical systems and Markov chains, and derives Edgeworth expansions, using a novel random complex Ruelle-Perron-Frobenius approach.

Contribution

It introduces a new method based on a random complex Ruelle-Perron-Frobenius theorem to analyze asymptotic moments and Edgeworth expansions in random dynamical environments.

Findings

Asymptotic moments exist and are computable via derivatives of a pressure function.

Asymptotic moments satisfy relations similar to those of i.i.d. sums.

Edgeworth expansions are derived for these processes.

Abstract

We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at the $0$ of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $\gam_k=\lim_{n\to\infty}n^{-[\frac k2]}\bbE(\sum_{i=1}^n X_i)^k$ of sums of independent and identically distributed random variables satisfy. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs rely on a (parametric) random complex Ruelle-Perron-Frobenius theorem, which replaces some the spectral techniques which are used in literature in order to obtain limit theorems for deterministic dynamical systems and Markov chains.

Equations

$${\gamma}_{k}=\lim_{n\to\infty}n^{-[\frac{k}{2}]}{\mathbb{E}}\big{(}\sum_{j=1}^{n}X_{j}\big{)}^{k}$$

$$\sup_{s\in{\mathbb{R}}}\big{|}\sqrt{2\pi}P(\sum_{j=1}^{n}X_{j}\leq{\sigma}\sqrt{n}s)-\int_{-\infty}^{s}e^{-\frac{t^{2}}{2}}dt-\sum_{j=1}^{d}n^{-\frac{j}{2}}P_{j}(s)e^{-\frac{s^{2}}{2}}\big{|}=o(n^{-\frac{d}{2}})$$

$${\mu }_{\omega }(e^{z{S}_n^{\omega }})={\mu }_{{\theta }^n\omega }({\mathcal{A}}_z^{\omega ,n}\text{1})$$

$${\Vert ({\lambda }_{\omega ,n}(z){)}^{-1}{\mathcal{A}}_z^{\omega ,n}g-{\nu }_{\omega }(z)(g)h_{{\theta }^n\omega }(z)\Vert }_1\le C\Vert g{\Vert }_1{\delta }^n$$

$${\gamma }_k(\omega )=\underset{n\to \infty }{\lim }{\gamma }_{k,n}(\omega )=\underset{n\to \infty }{\lim }{\gamma }_{k,n}({\theta }^{-n}\omega )$$

$${\int }_{{\mathcal{E}}_{\omega }}(S_n^{\omega }(x){)}^{k}d{\mu }_{\omega }(x)={\mu }_{{\theta }^n\omega }(h_{{\theta }^n\omega }^{(j)}(0))+\sum _{s=1}^{[\frac{k}{2}]}{C}_{\omega ,n}^{(k)}(s)n^s+{\epsilon }_{\omega ,n}$$

$$\underset{n\to \infty }{\lim }{C}_{\omega ,n}^{(k)}([\frac{k}{2}])=\underset{n\to \infty }{\lim }{C}_{{\theta }^{-n}\omega ,n}^{(k)}([\frac{k}{2}])={\gamma }_k(\omega )$$

$$\underset{n\to \infty }{\lim }{C}_{{\theta }^{-n}\omega ,n}^{(k)}(s)=$$

$$\displaystyle\sum_{j=0}^{k}j!\binom{k}{j}\mu_{\omega}(h_{\omega}^{(k-j)}(0))\sum_{m_{2},...,m_{j}}\big{(}\prod_{l=2}^{j}(i!)^{m_{l}}m_{l}!\big{)}^{-1}\prod_{l=2}^{j}({\mathbb{E}}_{P}[\Pi_{\omega}^{(l)}|{\mathcal{I}}])^{m_{l}}.$$

$$\underset{n\to \infty }{\lim }{n}^{r+\frac{1}{2}}\underset{t\in J}{\sup }\Vert {\mathcal{A}}_{it}^{\omega ,n}{\Vert }_1=0.$$

$$\underset{t:K_{\omega }\le |t|\le D_{\omega }{n}^r}{\sup }\Vert {\mathcal{A}}_{it}^{\omega ,n}{\Vert }_2\le C_{\omega }{n}^{-r_2(\omega )}.$$

$$\displaystyle\lim_{n\to\infty}n^{\frac{1}{2}}\sup_{s\in{\mathbb{R}}}\Big{|}\sqrt{2\pi}\mu_{\omega}\{x\in{\mathcal{E}}_{\omega}:S_{n}^{\omega}(x)\leq\sqrt{n}s\}-$$

$$\displaystyle\frac{1}{\sqrt{\Pi_{{\omega},n,2}}}\int_{-\infty}^{s}e^{-\frac{t^{2}}{2\Pi_{{\omega},n,2}}}dt-n^{-\frac{1}{2}}P_{{\omega},n,1}(s)e^{-\frac{s^{2}}{2}}\Big{|}=0.$$

$$\displaystyle\lim_{n\to\infty}n^{\frac{d}{2}}\sup_{s\in{\mathbb{R}}}\Big{|}\sqrt{2\pi}\mu_{\omega}\{x\in{\mathcal{E}}_{\omega}:S_{n}^{\omega}(x)\leq\sqrt{n}s\}-$$

$$\displaystyle\frac{1}{\sqrt{\Pi_{{\omega},n,2}}}\int_{-\infty}^{s}e^{-\frac{t^{2}}{2\Pi_{{\omega},n,2}}}dt-\sum_{j=1}^{d}n^{-\frac{j}{2}}P_{{\omega},n,j}(s)e^{-\frac{s^{2}}{2}}\Big{|}=0$$

$$\left|n^{-[\frac{k}{2}]}{\int }_{{\mathcal{E}}_{\omega }}(S_n^{\omega }(x){)}^{k}d{\mu }_{\omega }(x)-{\gamma }_k\right|\le R_{\omega ,k}{n}^{-\frac{1}{2}}\ln n$$

$$\underset{s\in \mathbb{R}}{\sup }\left|{\mu }_{\omega }\{x\in {\mathcal{E}}_{\omega }:S_n^{\omega }(x)\le s\sqrt{n}\}-\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\infty }^s{e}^{-\frac{t^2}{2{\sigma }^2}}dt\right|\le c_{\omega }{n}^{-\frac{1}{2}}\ln n.$$

$${\nu }_{\omega }(z)(h_{\omega }(z))=1,{\mathcal{A}}_z^{\omega }{h}_{\omega }(z)={\lambda }_{\omega }(z)h_{\theta \omega }(z)\text{ and }({\mathcal{A}}_z^{\omega }{)}^{\ast }{\nu }_{\theta \omega }(z)={\lambda }_{\omega }(z){\nu }_{\omega }(z).$$

$${\mathcal{A}}_z^{\omega }{h}_{\omega }(z)=\underset{n\to \infty }{\lim }({\lambda }_{{\theta }^{-n}\omega ,n}(z){)}^{-1}{\mathcal{A}}_z^{\omega }({\mathcal{A}}_z^{{\theta }^{-n}\omega ,n}\text{1})$$

$$={\lambda }_{\omega }(z)\underset{n\to \infty }{\lim }({\lambda }_{{\theta }^{-(n+1)}\theta \omega ,n+1}(z){)}^{-1}{\mathcal{A}}_z^{{\theta }^{-(n+1)}\theta \omega ,n+1}={\lambda }_{\omega }(z)h_{\theta \omega }(z).$$

$${\nu }_{\theta \omega }(z)({\mathcal{A}}_z^{\omega }g)=\underset{n\to \infty }{\lim }\frac{{\mathcal{A}}_z^{\theta \omega ,n}({\mathcal{A}}_z^{\omega }g)}{{\lambda }_{\theta \omega ,n}(z)h_{{\theta }^{n+1}\omega }(z)}$$

$$={\lambda }_{\omega }(z)\underset{n\to \infty }{\lim }\frac{{\mathcal{A}}_z^{\omega ,n+1}g}{{\lambda }_{\omega ,n+1}(z)h_{{\theta }^{n+1}\omega }(z)}={\lambda }_{\omega }(z){\nu }_{\omega }(z)(g)$$

$$h_{\omega }(z)=\underset{n\to \infty }{\lim }\frac{{\mathcal{A}}_z^{{\theta }^{-n}\omega ,n}\text{1}}{{\lambda }_{{\theta }^{-n}\omega ,n}(z)}$$

$${\lambda }_{{\theta }^{-n}\omega ,n}(z)={\lambda }_{{\theta }^{-n}\omega ,n}(z){\nu }_{{\theta }^{-n}\omega }(z)\text{1}={\nu }_{\omega }(z)({\mathcal{A}}_z^{{\theta }^{-n}\omega ,n}\text{1})$$

$${\lambda}_{{\omega},n}^{\prime}(0)=\nu_{{\theta}^{n}{\omega}}^{\prime}(0)\textbf{1}+\frac{d\nu_{{\theta}^{n}{\omega}}(0)({\mathcal{A}}_{z}^{{\omega},n}\textbf{1})}{dz}\Big{|}_{z=0}=\frac{d\mu_{\omega}(e^{zS_{n}^{\omega}})}{dz}\Big{|}_{z=0}=\mu_{\omega}(S_{n}^{\omega})=0.$$

$$\mu_{\omega}(e^{zS_{n}^{\omega}})=\mu_{{\theta}^{n}{\omega}}({\mathcal{A}}_{z}^{{\omega},n}\textbf{1})={\lambda}_{{\omega},n}(z)\big{(}\mu_{{\theta}^{n}{\omega}}(h_{{\theta}^{n}{\omega}}(z))+{\delta}_{{\omega},n}(z)\big{)}$$

$${\mu }_{\omega }(S_n^{\omega }{)}^k=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\lambda }_{\omega ,n}^{(j)}(0){\mu }_{{\theta }^n\omega }(h_{{\theta }^n\omega }^{(k-j)}(0))+d_{\omega ,n}$$

$${\lambda}_{{\omega},n}^{(j)}(0)=j!\sum_{s=1}^{[\frac{j}{2}]}n^{s}\sum_{(m_{2},...,m_{k})\in{\Gamma}_{j,s}}\big{(}\prod_{l=2}^{j}(l!)^{m_{l}}m_{l}!\big{)}^{-1}\prod_{l=2}^{j}\Big{(}n^{-1}\sum_{i=0}^{n-1}\Pi_{{\theta}^{i}{\omega}}^{(l)}(0)\Big{)}^{m_{l}}$$

$$\sum _{j=0}^{n-1}{\Pi }_{{\theta }^j\omega }^{\prime }(0)={\lambda }_{\omega ,n}^{\prime }(0)={\mu }_{\omega }(S_n^{\omega })=0.$$

$$\left|\frac{\mu_{\omega}(S_{n}^{\omega})^{k}}{n^{[\frac{k}{2}]}}-C_{k}\Big{(}n^{-1}\sum_{i=0}^{n-1}\Pi_{{\theta}^{i}{\omega}}^{(2)}(0)\Big{)}^{\frac{k}{2}}\right|\leq R_{k}n^{-1}$$

$${\mu }_{\omega }(h_{\omega }^{\prime }(0))=-{\nu }_{\omega }^{\prime }(0)\text{1}=0$$

$$\left|\frac{\mu_{\omega}(S_{n}^{\omega})^{k}}{n^{[\frac{k}{2}]}}-D_{k}\Big{(}n^{-1}\sum_{i=0}^{n-1}\Pi_{{\theta}^{i}{\omega}}^{(2)}(0)\Big{)}^{\frac{k-3}{2}}\cdot\Big{(}n^{-1}\sum_{i=0}^{n-1}\Pi_{{\theta}^{i}{\omega}}^{(3)}(0)\Big{)}\right|\leq R_{k}n^{-1}.$$

$$\Big{\|}\nu_{{\omega}}(z)(g)-\frac{{\mathcal{A}}^{{\omega},n}_{z}g}{{\mathcal{A}}^{{\omega},n}_{z}\textbf{1}}\Big{\|}_{1}\leq A{\delta}^{n}$$

Full text

1. Introduction

Probabilistic limit theorems for dynamical systems and Markov chains is a well studied topic. One way to derive such results is relying on some quasi-compactness (or spectral gap) of an appropriate transfer or Markov operator, together with a suitable perturbation theorem (see [28], [29], [13] and [20]). This quasi-compactness can often be verified only via an appropriate Ruelle-Perron-Frobenius (RPF) theorem, which is the main key for thermodynamic formalism type constructions. Probabilistic limit theorems for random dynamical systems and Markov chains in random dynamical environments were also studied in the literature (see, for instance, [21], [22], [14], [2] ,[8] and references therein). In these circumstances, the probabilistic behaviour of the appropriate process is determined by compositions of random operators, and not of a single operator, so no spectral theory can be exploited, and instead, many of these results rely on an appropriate version of the RPF theorem for random operators. For instance, the large deviations theorem in [21] rely on such results for real operators, and the conditions guaranteeing that the central limit theorem in [22] holds true can be often verified by an RPF theorem for random real operators. Relying on certain contraction properties of random complex transfer and Markov operators, with respect to a complex version of the Hilbert projective metric due to H.H. Rugh [30] (see also [10] and [11]), we proved in [14] an RPF theorem for random complex operators and presented the appropriate random complex thermodynamic formalism type constructions, which was one of the main keys in the proof of versions of the Berry-Esseen theorem and the local central limit theorem (LCLT) for certain processes in random dynamical environment, and in the proof of some nonconventional LCLT for dynamical systems (see Chapters 2 and 7 of [14]). In this paper we use this RPF theorem in order to derive additional limit theorems, as described in the following paragraphs.

Let $({\Omega},{\mathcal{F}},P,{\theta})$ be an invertible measure preserving system, let ${\mathcal{X}}$ be a compact metric space, ${\mathcal{E}}_{\omega}\subset{\mathcal{X}}$ be a measurable family of compact subsets and $u_{\omega}:{\mathcal{E}}_{\omega}\to{\mathbb{R}}$ be a random function, where ${\omega}\in{\Omega}$. Let $T_{\omega}:{\mathcal{E}}_{\omega}\to{\mathcal{E}}_{{\theta}{\omega}}$ be a random distance expanding map and let $\xi_{0}^{{\omega}},\xi_{1}^{{\theta}{\omega}},\xi_{2}^{{\theta}^{2}{\omega}},...$ be a Markov chain in the random dynamical environment $({\Omega},{\mathcal{F}},P,{\theta})$. In this paper we consider sequences of random variables having either the form $S_{n}^{\omega}=\sum_{j=0}^{n-1}u_{{\theta}^{j}{\omega}}\circ T_{\omega}^{j}(x)$ or the form ${\mathcal{S}}_{n}^{\omega}=\sum_{j=0}^{n-1}u_{{\theta}^{j}{\omega}}(\xi_{j}^{{\theta}^{j}{\omega}})$, where $T_{\omega}^{j}=T_{{\theta}^{j-1}{\omega}}\circ\cdots\circ T_{{\theta}{\omega}}\circ T_{\omega}$ and $x$ is distributed according to a special (Gibbs) measure $\mu_{\omega}$. Recall the following result. Let $X_{1},X_{2},X_{3},...$ be a sequence of centered, independent and identically distributed random variables with finite moments of all orders, and set ${\sigma}^{2}={\mathbb{E}}X_{1}^{2}$ and $\zeta={\mathbb{E}}X_{1}^{3}$. Then for each integer $k\geq 2$ the limit

[TABLE]

of the $k$-th normalized moment exists. Moreover, ${\gamma}_{2}={\sigma}^{2}$ and ${\gamma}_{3}=\zeta$, and when $k$ is even we have ${\gamma}_{k}=C_{k}{\sigma}^{k}$, where $C_{k}=2^{-\frac{k}{2}}(\frac{k}{2}!)^{-1}k!$, while for odd $k$’s larger than $2$ we have ${\gamma}_{k}=D_{k}{\sigma}^{k-3}\zeta$, where $D_{k}=\frac{k!}{3!}2^{-\frac{1}{2}(k-3)}(\frac{k-3}{2}!)^{-1}$. This result is a direct consequence of the multinomial theorem, and we refer the readers to Lemma 4.1 and Corollary 4.2 in [25] for a generalization to several weakly dependent sequences of random variables. In this paper, we will prove generalize these results for the sequences $S_{n}^{\omega}$ and ${\mathcal{S}}_{n}^{\omega}$ defined above by showing that the $k$-th normalized moments of the random variables $S_{n}^{\omega}$ and ${\mathcal{S}}_{n}^{\omega}$ behave like powers of ergodic averages of the second and third derivative at [math] of some pressure function, which will imply that the asymptotic moments exist, satisfy the desired relations and that they do not depend on ${\omega}$ when $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic. The fact that the second (normalzied) moments behave like an ergodic average was proved in [22] in a more general setup, and here we show that this is true (in our context) also for all the higher moments.

When $({\Omega},{\mathcal{F}},P,{\theta})$ satisfies some mixing conditions then we obtain almost sure convergence rate of order $n^{-\frac{1}{2}}\ln n$ towards the asymptotic moments, which allows us to obtain almost optimal convergence rate (of the same order) in the corresponding quenched central limit theorem for the sequences $n^{-\frac{1}{2}}S_{n}^{\omega}$ and $n^{-\frac{1}{2}}{\mathcal{S}}_{n}^{\omega}$. The arguments in the proof of the above convergence rate also yields that for sufficiently well mixing base maps ${\theta}$ the partial sums generated by the random RPF triplets (corresponding to an appropriate parametric family of random transfer operators) and all of their parametric derivatives satisfy several probabilistic limit theorems (see Remark 3.2), which also seems to be a new result.

Let $X_{1},X_{2},X_{3},...$ be a sequence of centered iid random variables with finite moments. Then, under certain assumptions on the behavior of the characteristic function of $X_{1}$, a classical result (see [12]) states that (Edgeworth) expansions of the form

[TABLE]

hold true for some polynomials $P_{1},P_{2},...,P_{d}$. Edgeworth expansion have been obtained in [29] for some classes of Markov chains and in [6] and [25] for deterministic dynamical systems (and Markov chains), and in this paper we will also obtain Edgeworth type expansions for the processes $S_{n}^{\omega}$ and ${\mathcal{S}}_{n}^{\omega}$ described earlier, with random polynomials $P_{i}=P_{i,{\omega},n}$ whose degree depends only on $i$, and the coefficients of $P_{i,{\theta}^{-n}{\omega},n}$ converge as $n\to\infty$. When the dynamical environment $({\Omega},{\mathcal{F}},P,{\theta})$ satisfies certain mixing conditions we also obtain certain convergence rate in the later limits. In general, in order to obtain Edgeworth type expansions by analysing the asymptotic behaviour of the characteristic functions $\varphi_{n}(t)={\mathbb{E}}e^{itS_{n}}$ of an underlying sequence $S_{n},\,n\geq 1$ of random variables, it is also necessary to have estimates on the decay of these functions (as $n\to\infty$) for large $t$’s, which is beyond the applications of the random complex thermodynamic formalism (the decay estimates sufficient for the existence of Edgeworth expansions in the deterministic case are discussed in [6] and [25]). Here ${\mathbb{E}}$ denotes expectation with respect to $P$. In the deterministic case, some of these estimates were obtained using the spectral theory of appropriate perturbations of the original transfer or Markov operator. In the circumstances of this paper there is no single operator, but a random family of operators, and we will use the approach taken in [14] in order to obtain the appropriate estimates in the case of iterates of random operators.

The paper is organized as follows. In Section 2 we formulate our results for random variables $S_{n}^{\omega}$ whose characteristic function is related to certain random operators ${\mathcal{A}}_{it}^{\omega},\,t\in{\mathbb{R}}$ which satisfy some complex thermodynamic formalism type assumption, together with certain decay rates of the norms of the iterates ${\mathcal{A}}_{it}^{{\omega},n}={\mathcal{A}}_{it}^{{\theta}^{n-1}{\omega}}\circ{\mathcal{A}}_{it}^{{\theta}^{n-2}{\omega}}\circ\cdots\circ{\mathcal{A}}_{it}^{\omega}$ of these operators for large $t$’s. In Section 3 we prove the results concerning the normalized asymptotic moments and in Section 4 we prove the results concerning Edgeworth expansions. Section 5 is devoted to examples of random dynamical systems $T_{\omega}$ and Markov chains $\xi_{0}^{\omega},\xi_{1}^{{\theta}{\omega}},\xi_{2}^{{\theta}^{2}{\omega}},...$ in random environments satisfying our assumptions.

2. Preliminaries and main results

Our setup consists of a complete probability space $({\Omega},{\mathcal{F}},P)$ together with an invertible $P$-preserving transformation ${\theta}:{\Omega}\to{\Omega}$, of a compact metric space $({\mathcal{X}},\rho)$ normalized in size so that $\text{diam}{\mathcal{X}}\leq 1$ together with the Borel ${\sigma}$-algebra ${\mathcal{B}}$, and of a set ${\mathcal{E}}\subset{\Omega}\times{\mathcal{X}}$ measurable with respect to the product ${\sigma}$-algebra ${\mathcal{F}}\times{\mathcal{B}}$ such that the fibers ${\mathcal{E}}_{\omega}=\{x\in{\mathcal{X}}:\,({\omega},x)\in{\mathcal{E}}\},\,{\omega}\in{\Omega}$ are compact. The latter yields (see [7] Chapter III) that the mapping ${\omega}\to{\mathcal{E}}_{\omega}$ is measurable with respect to the Borel ${\sigma}$-algebra induced by the Hausdorff topology on the space ${\mathcal{K}}({\mathcal{X}})$ of compact subspaces of ${\mathcal{X}}$ and the distance function $\rho(x,{\mathcal{E}}_{\omega})$ is measurable in ${\omega}$ for each $x\in{\mathcal{X}}$. Furthermore, the projection map $\pi_{\Omega}({\omega},x)={\omega}$ is measurable and it maps any ${\mathcal{F}}\times{\mathcal{B}}$-measurable set to an ${\mathcal{F}}$-measurable set (see “measurable projection” Theorem III.23 in [7]).

Let $(H_{1}^{\omega},\|\cdot\|_{1})$ be a Banach space of complex valued functions on ${\mathcal{E}}_{\omega}$, containing the constant functions, so that $\|g\|_{1}\geq\sup|g|$ for any $g\in H_{1}^{\omega}$, and $\|\textbf{1}\|_{1}=1$, where 1 denotes the function taking the constant value $1$ (regardless of its domain). Denote by $(H_{1}^{\omega})^{*}$ the space of all continuous linear functionals on $(H_{1}^{\omega},\|\cdot\|_{1})$, equipped with the operator norm $\|\cdot\|_{1}$. Let ${\mathcal{A}}_{z}^{\omega}:H_{1}^{\omega}\to H_{1}^{{\theta}{\omega}},\,z\in{\mathbb{C}}$ be a family of continuous linear operators, $S_{n}^{\omega}:{\mathcal{E}}_{\omega}\to{\mathbb{R}},\,n\geq 1$ be a sequence of Borel measurable functions and $\mu_{\omega}$ be a random probability measure on ${\mathcal{E}}_{\omega}$ so that $P$-a.s. we have $\mu_{\omega}(S_{n}^{\omega})=0$, $\|S_{n}^{\omega}\|_{L^{k}({\mathcal{E}}_{\omega},\mu_{\omega})}<\infty$ for any $k\geq 1$ and for any $z\in{\mathbb{C}}$ and $n\geq 1$,

[TABLE]

where ${\mathcal{A}}_{z}^{{\omega},n}={\mathcal{A}}_{z}^{{\theta}^{n-1}{\omega}}\circ{\mathcal{A}}_{z}^{{\theta}^{n-2}{\omega}}\circ\cdots\circ{\mathcal{A}}_{z}^{\omega}$. In this paper we will study certain (asymptotic) properties the distribution of $S_{n}^{\omega}(x)$, when $x$ is distributed according to $\mu_{\omega}$, and ${\omega}$ ranges over a set of full $P$-probability. Our basic requirements from the operators are described in the following

2.1 Assumption**.**

There exist constants $\rho,C>0$ and ${\delta}\in(0,1)$ so that $P$-a.s. for any $z\in B(0,\rho):=\{\zeta\in{\mathbb{C}}:|\zeta|<\rho\}$ there is a triplet consisting of a nonzero complex number ${\lambda}_{\omega}(z)$, a function $h_{\omega}(z)\in H_{1}^{\omega}$ and a linear functional $\nu_{\omega}(z)\in(H_{1}^{\omega})^{*}$ such that $\nu_{\omega}(z)\textbf{1}=1$ and for any $g\in H_{1}^{\omega}$ and $n\geq 1$,

[TABLE]

where ${\lambda}_{{\omega},n}(z)=\prod_{j=0}^{n-1}{\lambda}_{{\theta}^{j}{\omega}}(z)$. Moreover, the above triplet is measurable in ${\omega}$, analytic in $z$ and the random variables $\sup_{z:|z|<\rho}|{\lambda}_{\omega}(z)|$, $\sup_{z:|z|<\rho}\|h_{\omega}(z)\|_{1}$ and $\sup_{z:|z|<\rho}\|\nu_{\omega}(z)\|_{1}$ are bounded. Furthermore, $\nu_{\omega}(0)=\mu_{\omega}$, $h_{\omega}(0)=\textbf{1}$ and ${\lambda}_{\omega}(0)=1$.

We refer the readers’ to Section 5 for several examples in which Assumption 2.1 holds true. Under Assumption 2.1 we have $\nu_{\omega}(z)(h_{\omega}(z))=1$, ${\mathcal{A}}_{z}^{\omega}h_{\omega}(z)={\lambda}_{\omega}(z)h_{{\theta}{\omega}}(z)$ and $({\mathcal{A}}_{z}^{\omega})^{*}\nu_{{\theta}{\omega}}(z)={\lambda}_{\omega}(z)\nu_{\omega}(z)$ (see the beginning of Section 3), where $({\mathcal{A}}_{z}^{\omega})^{*}:(H_{1}^{{\theta}{\omega}})^{*}\to(H_{1}^{\omega})^{*}$ is the dual operator of ${\mathcal{A}}_{z}^{\omega}$ (i.e. the triplet from the assumption is a random complex Ruelle-Perron-Frobenius triplet, see Chapter 4 in [14]). In particular ${\mathcal{A}}_{0}^{\omega}\textbf{1}=\textbf{1}$.

Our first result is the following

2.2 Theorem**.**

(i) By possibly decreasing $\rho$, $P$-a.s. we can define an analytic function $\Pi_{\omega}:B(0,\rho)\to{\mathbb{C}}$ so that $\Pi_{\omega}(0)=0$, ${\lambda}_{\omega}(z)=e^{\Pi_{\omega}(z)}$ and $|\Pi_{\omega}(z)|\leq c_{0}$ for any $z\in B(0,\rho)$, for some constant $c_{0}$ which does not depend on ${\omega}$ and $z$.

(ii) Suppose that $\int S_{n}^{\omega}\mu_{\omega}=0$ for $P$-almost any ${\omega}$. For any $k\geq 2$ and $n\geq 1$, let ${\gamma}_{k,n}$ be the random variable defined by ${\gamma}_{k,n}({\omega})=n^{-[\frac{k}{2}]}\int_{{\mathcal{E}}_{\omega}}(S_{n}^{\omega}(x))^{k}d\mu_{\omega}(x)$. Then, the limits

[TABLE]

exist $P$-a.s. and in $L^{p}$, for any $1\leq p<\infty$. Moreover, with ${\sigma}^{2}={\gamma}_{2}$, $\zeta={\gamma}_{3}$, $C_{k}=2^{-\frac{k}{2}}(\frac{k}{2}!)^{-1}k!$ and $D_{k}=\frac{k!}{3!}2^{-\frac{1}{2}(k-3)}(\frac{k-3}{2}!)^{-1}$, when $k$ is even we have ${\gamma}_{k}=C_{k}{\sigma}^{k}$, while for odd $k$’s we have ${\gamma}_{k}=D_{k}{\sigma}^{k-3}\zeta$. Furthermore, ${\sigma}^{2}={\gamma}_{2}={\mathbb{E}}[\Pi_{\omega}^{\prime\prime}(0)|{\mathcal{I}}]$ and $\zeta={\gamma}_{3}={\mathbb{E}}[\Pi_{\omega}^{\prime\prime\prime}(0)|{\mathcal{I}}]$, where ${\mathcal{I}}$ is the sub-$\sigma$ algebra of ${\mathcal{F}}$ containing only the ${\theta}$-invariant sets. In particular, when the measure preserving system $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic then all ${\gamma}_{k}({\omega})$’s do not depend on ${\omega}$.

(iii) In fact, we can write

[TABLE]

where $|{\varepsilon}_{{\omega},n}|\leq c_{1}\eta^{n}$ for some constants $c_{1}>0$ and $0<\eta<1$,

[TABLE]

$C_{{\omega},n}^{(k)}([\frac{k}{2}]-1)=C_{{\omega},n}^{(k)}(1)=0$* and for all other $s$’s,*

[TABLE]

Here the sum ranges over all possible choices of nonnegative integers $m_{2},...,m_{j}$ so that $\sum_{l=2}^{j}m_{l}=s$ and $\sum_{l=2}^{j}lm_{l}=j$, and $f^{(l)}$ stands for the $l$-th derivatives of a function $f$ on $B(0,\rho)$ which takes values at some Banach space. Moreover, closed formulas for the $C_{{\omega},n}^{(k)}$’s can be recovered from the proof.

As in [25], the method used in the proof of Theorem 2.2 yields that the limits ${\gamma}_{k}$ exist also without the assumption that $\int S_{n}^{\omega}d\mu_{\omega}=0$, but then the relations between the ${\gamma}_{k}$’s are more complicated and will not be discussed in this paper.

Now we will describe our results concerning Edgeworth type expansions. For any $r>0$ consider the following assumptions

2.3 Assumption**.**

(i) There exists a random variable $a_{\omega}$ so that $P$-a.s. we have $\|{\mathcal{A}}_{it}^{{\omega},n}\|_{1}\leq a_{\omega}$ for any $t\in(-\rho,\rho)$, where $\rho$ comes from Assumption 2.1.

(ii) For any compact set $J\subset{\mathbb{R}}\setminus\{0\}$, $P$-.a.s for any $n\geq 1$ we have

[TABLE]

2.4 Assumption**.**

There exists a Banach space $(H_{2}^{\omega},\|\cdot\|_{2})$ containing the constant functions, such that $\|g\|_{2}\geq\sup|g|$ for any $g\in H_{2}^{\omega}$, and ${\mathcal{A}}_{it}^{\omega}$, $t\in{\mathbb{R}}$, can be (possibly) extended to $H_{2}^{\omega}$ so that ${\mathcal{A}}_{it}$ is a continuous map between $H_{2}^{\omega}$ and $H_{2}^{{\theta}{\omega}}$. Moreover:

(i) There exists a random variable $R_{\omega}$ so that $P$-a.s. for any $t\in{\mathbb{R}}$ and $n\geq 1$ we have $\|{\mathcal{A}}_{it}^{{\omega},n}\|_{2}\leq R_{\omega}(1+|t|)$.

(ii) There exist random variables $K_{\omega},D_{\omega},C_{\omega}>0$, $r_{2}({\omega})>0$ and $N({\omega})$ so that $P$-a.s. for any $n\geq N({\omega})$,

[TABLE]

2.5 Theorem**.**

Suppose that Assumptions 2.1 and 2.3 hold true with $r=\frac{1}{2}$. Set $\Pi_{{\omega},n,2}=n^{-1}\sum_{j=0}^{n-1}\Pi_{{\theta}^{j}{\omega}}^{\prime\prime}(0)$, and assume that ${\sigma}^{2}=\lim_{n\to\infty}\Pi_{{\omega},n,2}=\lim_{n\to\infty}\Pi_{{\theta}^{-n}{\omega},n,2}={\mathbb{E}}[\Pi_{\omega}^{\prime\prime}(0)]|{\mathcal{I}}]>0$.

(i) There exists a sequence of polynomials $P_{{\omega},n,1}(s)=\sum_{j=0}^{m_{1}}a_{{\omega},n,j,1}s^{j},\,n\geq 1$ with random coefficients, whose degree $m_{1}$ does not depend on ${\omega}$ and $n$, so that $P$-almost surely we have

[TABLE]

(ii) Let $d\in{\mathbb{N}}$. If Assumptions 2.3 and 2.4 hold true with $r=\frac{d}{2}$ and $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic, then for each $k\geq 2$ there exists a sequences of polynomials $P_{{\omega},n,k}(s)=\sum_{j=0}^{m_{k}}a_{{\omega},n,j,k}s^{j},\,n\geq 1$ with random coefficients and degree $m_{k}$ which depends only on $k$, so that $P$-almost surely we have

[TABLE]

(iii) The coefficients $a_{{\omega},n,j,k}$ of the above polynomials are algebraic combinations of the derivatives of the functions $\mu_{{\theta}^{n}{\omega}}(h_{\omega}(z))$ and $n^{-1}\sum_{j=0}^{n-1}\Pi_{{\theta}^{j}{\omega}}(z)$ at $z=0$. In particular, they are uniformly bounded in ${\omega}$ and $n$, where ${\omega}$ ranges over a set of probability $1$, and $a_{{\theta}^{-n}{\omega},n,j,k}$ converges $P$-a.s. and in $L^{p}$, for any $p\in[1,\infty)$, as $n\to\infty$ towards a limit $a_{{\omega},j,k}$ The coefficients $a_{{\omega},n,j,k}$ and the corresponding limits can be recovered from the proof.

Note that when (2.3) below holds true then we obtain a converge rate of order $n^{-\frac{1}{2}}\ln n$ for all of the coefficients of the $P_{{\theta}^{-n}{\omega},n,k}$’s.

2.6 Remark**.**

A natural question is whether Theorem 2.5 (or a version of it with $S_{n}^{{\theta}^{-n}{\omega}}$ instead of $S_{n}^{\omega}$) can be obtained with polynomials which depend only on ${\omega}$ and not on $n$ and with ${\sigma}^{2}$ in place of $\Pi_{{\omega},n,2}$. Similar arguments to the ones in Remark 1.1 from [25] show that replacing $P_{{\omega},n,k}$ with a polynomial $P_{{\omega},k}$ which does not depend on $k$ is possible only if $n^{\frac{k}{2}}|P_{{\omega},k}(x)-P_{{\theta},n,k}(x)|$ converges to [math] as $n\to\infty$ uniformly in $x$, which means that the only candidate for such polynomials are the polynomials $P_{{\omega},k}$ whose coefficients are the limits of the coefficients of the $P_{{\omega},n,k}$’s (or the $P_{{\theta}^{-n}{\omega},n,k}$’s in the “reverse version”). Proving such a result does not seem possible using only the converge of the coefficients of $P_{{\theta}^{-n}{\omega},n,k}$ even when ${\theta}$ satisfies some mixing conditions, since this convergence is derived from the convergence of certain ergodic averages, which, in general, is slower than $n^{-\frac{1}{2}}$ (even for independent summands). Note that even for Edgeworth expansions of order one (as well as for a Berry-Esseen theorem) it is not clear how to obtain the results with ${\sigma}^{2}$ in place of $\Pi_{{\omega},n,2}$, since this essentially requires better convergence rate of the variance of $n^{-\frac{1}{2}}S_{n}^{\omega}$ towards ${\sigma}^{2}$. The difference between this variances and ${\sigma}^{2}$ behaves as an ergodic average, which leads again to the same problem discussed above (concerning the rate of converges of ergodic averages).

2.1. Almost sure convergence rates under mixing conditions

When $({\Omega},{\mathcal{F}},P,{\theta})$ satisfies certain mixing assumptions, then, in Section 3.1 we obtain almost sure converges rate of the form

[TABLE]

where $R_{{\omega},k}$ is some random variable. Under certain mixing condition the authors of [19] obtained converges rates of order $(\ln n)^{\frac{3}{2}+{\delta}_{0}}n^{-\frac{1}{2}},\,{\delta}_{0}>0$ for $k=2$, and in our setup we obtain such rates for all $k$’s, with ${\delta}_{0}=-\frac{1}{2}$. Using (2.3) with $k=2$, when ${\sigma}^{2}>0$ we also derive in Section 3.1 almost optimal convergence rate in the central limit theorem of the form

[TABLE]

3. Asymptotic moments: Proof of Theorem 2.2

First, since ${\lambda}_{\omega}(0)=1$, ${\lambda}_{\omega}(\cdot)$ is an analytic function and $|{\lambda}_{\omega}(z)|$ is bounded uniformly in ${\omega}$ and $z$, where ${\omega}$ ranges over a set of probability $1$ and $z\in B(0,\rho)$, it is indeed possible to construct a function $\Pi_{\omega}(\cdot)$ which satisfies the conditions stated in Theorem 2.2 (i), namely, an appropriate branch of the logarithm of ${\lambda}_{\omega}(z)$ can be defined on some deterministic neighborhood of [math].

Next, we will show that $({\lambda}_{\omega}(z),h_{\omega}(z),\nu_{\omega}(z))$ is an RPF triplet, namely that $P$-a.s. for any $z\in B(0,\rho)$,

[TABLE]

Indeed, in order to prove the second equality in (3.1), plug in ${\theta}^{-n}{\omega}$ in place of ${\omega}$ in (2.1), and then choose $g=\textbf{1}$, in order to deduce that

[TABLE]

Next, applying (2.1) with ${\theta}{\omega}$ in place of ${\omega}$ and with ${\mathcal{A}}_{z}^{\omega}g$ in place of $g$ we derive that for any $g\in H_{1}^{\omega}$,

[TABLE]

which implies that $({\mathcal{A}}_{z}^{\omega})^{*}\nu_{{\theta}{\omega}}(z)={\lambda}_{\omega}(z)\nu_{\omega}(z)$. Note that the function $h_{\omega}(z)$ does not vanishes when $z$ lies in a deterministic neighborhood of [math] since $h_{\omega}(0)\equiv 1$, the random variable $\sup_{|z|<\rho}\|h_{\omega}(z)\|_{1}$ is bounded and the map $z\to h_{\omega}(z)$ is analytic. Finally, since

[TABLE]

and

[TABLE]

we obtain that $\nu_{\omega}(z)(h_{\omega}(z))=1$. ∎

Now we will start proving Theorem 2.2. First, it follows from (3.1) that ${\lambda}_{{\omega},n}(z)=\nu_{{\theta}^{n}{\omega}}(z)({\mathcal{A}}_{z}^{{\omega},n}\textbf{1})$, and so, since $\nu_{\omega}(z)\textbf{1}=1$ and $\mu_{\omega}(e^{zS_{n}^{\omega}})=\mu_{{\theta}^{n}{\omega}}({\mathcal{A}}_{z}^{{\omega},n}\textbf{1})$,

[TABLE]

Next, using Assumption 2.1 for any $n\geq 1$ we can write

[TABLE]

where $|{\delta}_{{\omega},n}(z)|\leq C{\delta}^{n}$. Since ${\delta}_{{\omega},n}(\cdot)$ is an analytic function, it follows from the Cauchy integral formula that for any $k\geq 1$ there exists a constant $Q_{k}$ so that $|{\delta}_{{\omega},n}^{(k)}(z)|\leq Q_{k}{\delta}^{n}$ for any $z\in B(0,\frac{\rho}{2})$. Since ${\lambda}_{{\omega},n}(0)=1$ and $\sup_{z\in B(0,\rho)}|{\lambda}_{\omega}(z)|$ is a bounded random variable, the analyticity of ${\lambda}_{\omega}(\cdot)$ implies that for any $k\geq 1$ we have $|{\lambda}_{{\omega},n}^{(k)}(0)|\leq H_{k}n^{k}$, for some constant $H_{k}$ which does not depend on $n$ and ${\omega}$. Therefore, differentiating $k$ times both sides of (3.2) at $z=0$ yields that

[TABLE]

for some random variable $d_{{\omega},n}$ so that $|d_{{\omega},n}|\leq Ac^{n}$ for some $A\geq 1$ and $c\in(0,1)$, which do not depend on ${\omega}$. Since ${\lambda}_{{\omega},n}(z)=e^{\sum_{j=0}^{n-1}\Pi_{{\theta}^{j}{\omega}}(z)}$ and $\Pi_{\omega}(0)=1$, it follows from the Faá di Bruno formula that for any $j$,

[TABLE]

where ${\Gamma}_{j,s}$ is the set of all $j-1$-tuples $(m_{2},...,m_{j})$ of nonnegative integers so that $\sum_{l}lm_{l}=j$ and $\sum_{l}m_{l}=s$, and we took into account that

[TABLE]

Observe that when $s=[\frac{j}{2}]$ and $j$ is even we have ${\Gamma}_{j,s}=\{(\frac{j}{2},0,0,...,0)\}$, while for odd $j$’s we have ${\Gamma}_{j,s}=\{(\frac{j-3}{2},1,0,...,0)\}$. Since $\Pi_{\omega}(z)$ is analytic in $z$ and uniformly bounded in ${\omega}$ and $z$ (where ${\omega}$ ranges over a set of probability $1$ and $z\in B(0,\rho)$), for each $j$ there exists a constant $L_{j}$ so that $\sup_{z\in B(0,\frac{\rho}{2})}|\Pi_{\omega}^{(j)}(z)|\leq L_{j}$ for $P$-almost any ${\omega}$. When $k$ is even, by considering the case when $j=k$, we conclude that there exist constants $R_{k},k\geq 2$ so that for even $k$’s we have

[TABLE]

where $C_{k}=2^{-\frac{k}{2}}(\frac{k}{2}!)^{-1}k!$. Now, suppose that $k$ is odd. Then the dominating terms come from considering $j=k$ and $j=k-1$. Differentiating both sides of the equality $\nu_{\omega}(z)(h_{\omega}(z))=1$ and plugging in $z=0$ we derive that

[TABLE]

where we used that $h_{\omega}(0)\equiv\textbf{1}$ and that $\nu_{\omega}(z)\textbf{1}=1$ for any $z$. Therefore, we actually only need to consider the case when $j=k$ also for odd $k$’s, and so, with $D_{k}=\frac{k!}{3!}2^{-\frac{1}{2}(k-3)}(\frac{k-3}{2}!)^{-1}$ for odd $k$’s large than $2$ we have

[TABLE]

Since the $L^{\infty}$-norms of $n^{-1}\sum_{i=0}^{n-1}\Pi_{{\theta}^{i}{\omega}}^{(l)}(0)$ are uniformly bounded in $n$ (for each $l$), all the positive integer powers of these averages converge almost surely and in $L^{p}$, for any $p\in[1,\infty)$, towards the appropriate power of $E[\Pi_{\omega}^{(l)}(0)|{\mathcal{I}}]$. Combining this with the above estimates we complete the proof of Theorem 2.2.

3.1. Convergence rates towards the asymptotic moments under mixing conditions

We assume here that ${\Omega}={\mathcal{Y}}^{z}$ for some measurable space ${\mathcal{Y}}$, and that $({\Omega},{\mathcal{F}},P,{\theta})$ is the shift system generated by a ${\mathcal{Y}}$-valued stationary process $\{\xi_{n}:\,n\in{\mathbb{Z}}\}$. Consider the situation when ${\mathcal{A}}_{z}^{\omega}$ depends only on the [math]-th coordinate of ${\omega}$. Then by (2.1), when $z$ lies in some deterministic neighborhood of the origin, the functional $\nu_{\omega}(z)$ can be approximated in $L^{\infty}$ exponentially fast in $n$ by functions of the first $n$-th coordinates. Indeed, since $h_{\omega}(0)=\textbf{1}$, $h_{\omega}(z)$ is analytic and $\sup_{z\in B(0,\rho)}\|h_{\omega}(z)\|_{1}$ is bounded, we derive that there exist constants $r_{0}>0$ and ${\delta}_{0}>0$ so that $P$-a.s. for any $z\in B(0,r_{0})$ we have $\inf|h_{\omega}(z)|>{\delta}_{0}$. Since $\nu_{\omega}(z)\textbf{1}=1$, it follows now from (2.1) that for any $z\in B(0,r_{0})$ and $g\in H_{1}^{\omega}$ so that $\|g\|_{1}\leq 1$,

[TABLE]

where we used that $\nu_{{\omega}}(z)(g)=\frac{\nu_{{\omega}}(z)(g)}{\nu_{{\omega}}(z)\textbf{1}}$. Here $A>0$ is some constant and ${\delta}$ comes from (2.1). By (3.1) we have ${\lambda}_{\omega}(z)=\nu_{\omega}(z)({\mathcal{A}}_{z}^{\omega}\textbf{1})$, and therefore $P$-a.s. for any $z\in B(0,r_{0})$ and $n\geq 1$ we have

[TABLE]

where $C>0$ is some constant, ${\omega}=\{{\omega}_{s}:\,s\in{\mathbb{Z}}\}$ and $q_{n,z}:{\mathcal{Y}}^{n}\to{\mathbb{C}}$ is a family of measurable functions which is uniformly bounded in ${\omega}_{0},{\omega}_{1},...,{\omega}_{n-1}$, $z$ and $n$. Since ${\lambda}_{\omega}(0)=1$ and $\sup_{z\in B(0,\rho)}|{\lambda}_{{\omega}}(z)|$ is a bounded random variable, there exist constants $a,b,\rho>0$ so that $P$-a.s. for any $z\in B(0,\rho)$ we have $|{\lambda}_{\omega}(z)|\in(a,b)$. Therefore, also $\Pi_{{\omega}}(z)$ can be approximated exponentially fast in the above sense when $z$ belongs to a deterministic neighborhood of [math]. Thus, using the identifications

[TABLE]

we derive that for any $k$ there exists a constant $A_{k}$ and a uniformly bounded family of functions $g_{k,n}:{\mathcal{Y}}^{n}\to{\mathbb{C}}$ so that for any $j\geq 0$ and $n\geq 1$,

[TABLE]

Next, recall that the $\phi$-mixing (dependence) coefficients associated with the sequence $\{\xi_{n}:n\in{\mathbb{Z}}\}$ are given by

[TABLE]

where ${\mathcal{F}}_{-\infty,s}$ is the ${\sigma}$-algebra generated by $\{\xi_{l},\,l\leq s\}$ and ${\mathcal{F}}_{s,\infty}$ is the ${\sigma}$-algebra generated by $\{\xi_{l}:l\geq s\}$ (for any integer $s$). Recall (see [5], Ch. 4) also that $\phi(n)$ can be written as

[TABLE]

We will obtain convergence rates towards the asymptotic moments under the assumption that

[TABLE]

Under this assumption, the next step of the proof of (2.3) is to approximate $\sum_{j=0}^{n-1}\Pi_{{\theta}^{j}{\omega}}^{(k)}(0)$ by martingales in the $L^{\infty}$ norm. The following arguments are classical and are given here for readers’ convenience. For each $m\geq 0$, let ${\mathcal{F}}_{0,m}$ be the ${\sigma}$-algebra generated by $\xi_{0},\xi_{1},...,\xi_{m}$. For any $k\geq 2$ and $r\geq 1$ set

[TABLE]

and let $M^{(r,k)}_{n}=\sum_{j=0}^{n-1}X_{j}^{(k,r)}$ be the martingale (w.r.t the filtration $\{{\mathcal{F}}_{0,n-1+r}:n\geq 0\}$) whose differences are given by

[TABLE]

Then by (3.8), similarly to the proof of Theorem 2.8 in [15], we have $\|X_{j}^{(r,k)}\|_{L^{\infty}}\leq C(1+r+S_{\phi})$ for any $j$, where $C$ is some constant, and, using also (3.7) we derive that

[TABLE]

where $Q_{k}$ is some constant. Applying the Azuma-Hoeffding inequality (see, for instance, page 33 in [26]) with the martingales $M_{n}^{(k,r)}$ and $-M_{n}^{(k,r)}$, we obtain that that for any $t\geq 0$,

[TABLE]

where ${\lambda}_{t}=\frac{t}{2nC(1+r+S_{\phi})}$, and in the third inequality we used the Markov inequality. Taking $r=r_{n}$ of logarithmic order in $n$, we derive that there exist constants $a_{k},C_{k},c_{k},d_{k}>0$ so that for any $n\geq 2$ and ${\varepsilon}>0$,

[TABLE]

By taking ${\varepsilon}=un^{-\frac{1}{2}}\ln n$ for a sufficiently large $u$, we derive from the Borel Cantelli Lemma that for any $k$, $P$-a.s for any sufficiently large $n$ we have

[TABLE]

where $b_{k}>0$ is some constant which depends only on $k$. It follows from (3.9), (3.5) and (3.6) that the convergence rate towards the asymptotic moments is at most of order $n^{-\frac{1}{2}}\ln n$ (i.e. that (2.3) holds true).

Next, the arguments in the proof of Theorem 7.1.1 in [14] show that when ${\sigma}^{2}>0$ then there exist random variables $c_{\omega}$ and $d_{\omega}$ so that $P$-a.s. for any $n\geq 1$,

[TABLE]

Therefore, under the above mixing conditions we derive almost optimal convergence rate in the central limit theorem of the form

[TABLE]

where $q_{\omega}$ is some random variable (which in general may not even be integrable).

3.1 Remark**.**

When $\{\xi_{n}:\,n\geq 1\}$ is a stationary geometrically ergodic Markov chain then $\phi(n)$ converges exponentially fast to [math] (see [5], Theorem 21.1), and in particular $S_{\phi}<\infty$. Such exponential convergence occurs also when $\xi_{n}$ has form $\xi_{n}(x)=\xi_{0}\circ S^{n}(x)$, where $S$ is a two sided topologically mixing subshift of finite type, $x$ is distributed according to some Gibbs measure (see [3]) and $\xi_{0}$ is a function of a finite number of coordinates. If $\{\zeta_{n}:n\geq 0\}$ is a one sided stationary process then we can define $\xi_{n}=(\xi_{n,j})_{j=-\infty}^{\infty}=(...,\zeta_{n-1},\zeta_{n},\zeta_{n+1},...)$, where $\zeta_{n}$ appears in the [math]-the coordinate. In these circumstances, we can write ${\omega}=({\omega}_{j})_{j\in{\mathbb{Z}}}$ and ${\omega}_{j}=({\omega}_{j,k})_{k=-\infty}^{\infty}$. When ${\mathcal{A}}_{z}^{\omega}$ depends only on ${\omega}_{0,0}$ then all the arguments from this section hold true if we assume that the sequence $\phi_{1}(n),\,n\geq 1$ given by

[TABLE]

is summable, where ${\mathcal{G}}_{-\infty,s}$ is the ${\sigma}$-algebra generated by $\{\zeta_{0},\zeta_{1},...,\zeta_{s}\}$ and ${\mathcal{G}}_{s,\infty}$ is the ${\sigma}$-algebra generated by $\{\zeta_{s},\zeta_{s+1},...\}$. Note also that the above results hold true also when $\xi_{n}$ has the form $\xi_{n}(x)=\xi_{0}\circ T^{n}(x)$, where $T$ is a Young tower (see [31] and [32]) whose tails decay to [math] sufficiently fast, $x$ is distributed according to an appropriate Gibbs measure and $\xi_{0}$ is a bounded random variables which is measurable with respect to the partition which defines the tower. Young towers do not seem to be $\phi$-mixing with respect to the natural filtrations generated by cylinder sets, but they satisfy certain type of “graded $\phi$-mixing” conditions which allows use to obtain the results above (see Section 7 in [18] for several mixing properties of Young towers which involve certain $\phi$-mixing coefficients).

3.2 Remark**.**

The argument above show that the RPF triplets ${\lambda}_{\omega}(z),h_{\omega}(z)$ and $\nu_{\omega}(z)$ and all their derivatives with respect to $z$ can be approximated by functions of the random variables $\xi_{0},\xi_{1},...,\xi_{n}$ (or $\xi_{-n},...,\xi_{0}$ in the case of $h_{\omega}(z)$) with an error term which converges to [math] exponentially fast when $n\to\infty$. Therefore, applying classical results from probability theory (for stationary and mixing processes) we obtain that for any real $t$ with a sufficiently small absolute value, the partial sums $\sum_{j=0}^{n-1}X_{i,k}({\theta}^{j}{\omega})$ generated by the random variables $X_{1,k}({\omega})={\lambda}_{\omega}^{(k)}(t)$, X_{2,k}({\omega})=\big{(}h_{\omega}^{(k)}(t)\big{)}(x) or X_{3,k}({\omega})=\big{(}\nu^{(k)}_{\omega}(t)\big{)}(g) satisfy exponential concentration inequalities, as well as various limit theorems such as the CLT. Here we assume that ${\mathcal{E}}_{\omega}={\mathcal{X}}$ does not depend on ${\omega}$, $x\in{\mathcal{X}}$ and $g$ is a Hölder continuous function on ${\mathcal{X}}$.

4. Edgeworth type expansions: Proof of Theorem 2.5

The proof of Theorem 2.5 relies on classical arguments involving Fourier transforms which were used successfully in the i.i.d. case and in the deterministic case (i.e. when $|{\Omega}|=1$), see, for instance, [12], [25] and references therein. Most of the arguments in the proof of Theorem 2.5 are modifications of the arguments in [25]. When the arguments are exactly as in [25], we will just refer the reader’s there. The main difficulty is to verify Assumptions 2.1, 2.3 and 2.4, and this is postponed to Section 5.

For reader’s convenience, we first will describe the main idea behind using Fourier transforms in order to derive Edgeworth expansions. Recall first that by the Esseen inequality (see [24], Ch. 4.1), there exists an absolute constant $A$ so that for any $T>0$, a distribution function $F$ and an integrable function $G:{\mathbb{R}}\to{\mathbb{R}}$ with bounded first derivative such that $\lim_{x\to\infty}G(x)=0$,

[TABLE]

where $f(t)=\int e^{itx}dF(x)$ and $g(t)=\int e^{itx}dG(x)=\hat{G^{\prime}}(-t)$ (here $\hat{g}$ is the Fourier transform of a function $g$). Let $F(s)=\mu_{\omega}\{x:\in{\mathcal{E}}_{\omega}:S_{n}^{\omega}\leq\sqrt{n}s\}$. The idea behind the proof is to find a function $H$ so that $\frac{|\hat{F}(t)-H(t)|}{|t|}$ will be sufficiently small on some interval $[-T,T]$, where $T$ is of order $n^{-\frac{d}{2}}$. In this case we could take $G(s)=\int_{-\infty}^{s}\hat{H}(x)dx$ (so $\hat{G}^{\prime}(-t)=H(t)$) and it will remain to verify that $G$ has the desired form. On intervals of the form $I_{n}({\delta})=[-{\delta}\sqrt{n},{\delta}\sqrt{n}]$ for small ${\delta}$’s this approximation is done using Assumption 2.1, while on $J_{n}({\delta},a)=[-a\sqrt{n},a\sqrt{n}]\setminus I_{n}({\delta})$ (for large $a$’s) we will use Assumption 2.3. When $d>1$ we will use Assumption 2.4 in order to obtain an appropriate estimate on $[-T,T]\setminus J_{n}(a,{\delta})$.

Let $d\geq 1$. The first step of the proof of Theorem 2.5 is as follows. The arguments preceding (3.3) show that for any $z\in B(0,\frac{\rho}{2})$ we can write

[TABLE]

where $W_{{\omega}}(z)=\nu_{{\omega}}(z)(h_{{\omega}}(z))$, $d_{{\omega},n}(0)=0$ and $|{\delta}_{{\omega},n}^{(k)}(z)|\leq C_{k}{\delta}^{n}$. Since ${\delta}_{{\omega},n}$ is analytic and ${\delta}_{{\omega},n}(0)=0$, it follows that $|{\delta}_{{\omega},n}(z)|\leq C^{\prime}_{0}|z|{\delta}^{n}$ for $z$’s in the above domain, where $C_{0}^{\prime}$ is some constant. Moreover, it follows from Assumption 2.3 (i) together with (2.1) that $P$-a.s. we have $\sup\{|{\lambda}_{{\omega},n}(it)|:\,t\in(-r,r),n\geq 1\}\leq C_{0}^{\prime\prime}c_{\omega}$, where $c_{\omega}$ is some random variable. Therefore, $P$-a.s. for any $|t|<\frac{\rho}{2}$ and $n\geq 1$,

[TABLE]

for some random variable $C_{\omega}$. Next, since $\Pi_{{\omega}}(0)=\Pi_{\omega}^{\prime}(0)=0$ and all the derivatives of $\Pi_{\omega}(z)$ are uniformly bounded in ${\omega}$, where ${\omega}$ ranges over a set of probability $1$, we can write

[TABLE]

where

[TABLE]

By (4.2) and (4.3) we have for $P$-a.a. ${\omega}$,

[TABLE]

Let $k\geq 0$. Then, by Theorem 2.5 (i), there exists a constant $B_{k}>0$ so that $P$-a.s. we have $\sup_{z\in B(0,\rho)}|\psi_{{\omega},n}(z)|\leq B_{k}$ for any $n\geq 1$. Observe also that $\psi_{{\omega},n}(0)=\psi_{{\omega},n}^{\prime}(0)=\psi_{{\omega},n}^{\prime\prime}(0)=0$. Therefore we can write $\psi_{{\omega},n}(z)=z^{2}\psi_{{\omega},n,d}(z)+|z|^{d+2}\tilde{\psi}_{{\omega},n,d}(z)$, where $\psi_{{\omega},n,d}$ is a polynomial of degree $d$ and $\tilde{\psi}_{{\omega},n,d}$ is an analytic function which vanishes at $z=0$ and is bounded in some deterministic neighborhood of the origin by some constant which does not depend on ${\omega}$ and $n$. Using the latter notations, we can write

[TABLE]

Since $\mu_{\omega}(h_{\omega}(0))$, $\mu_{\omega}$ is a probability measure, $h_{\omega}(z)$ is analytic in $z$ and $\sup_{z\in B(0,\rho)}\|h_{\omega}(z)\|_{1}$ is a bounded random variable, there exists a constant $0<r_{0}<\frac{\rho}{2}$ so that $P$-a.s. we can develop a branch of $\ln W_{{\omega}}(z)$ in $B(0,r_{0})$ which is analytic, uniformly bounded in ${\omega}$ (when ${\omega}$ ranges over a set of probability $1$) and takes the value $\ln W_{\omega}(0)=0$ at $z=0$. Next, we can write $W_{{\omega}}(it)=1+W_{{\omega},d}(t)+t^{d}\tilde{W}_{{\omega},d}(t)$, where $W_{{\omega},d}(t)$ is a polynomial of degree $d$ which vanishes at $t=0$, whose coefficients are bounded random variables, and $\tilde{W}_{{\omega},d}(t)$ is a $C^{\infty}$ function which vanishes at $t=0$, whose derivatives are uniformly bounded in ${\omega}$ (around [math]). Similarly to the derivation of equality (3.7) in [25], and using the Taylor expansions of the functions $\exp(\cdot)$ and $\ln(\cdot)$ we derive that $P$-a.s. for any $t\in(-n^{\frac{1}{2}}r_{0},n^{\frac{1}{2}}r_{0})$,

[TABLE]

Here $A_{0}\equiv 1$ and all the other $A_{{\omega},n,k}$’s are polynomials of degree $s_{k}$ which does not depend on ${\omega}$ and $n$, whose coefficients are algebraic combinations of the derivatives of $W_{{\omega},n}$ at [math] and the derivatives of $\psi_{{\omega},n}$ at [math], and $\varphi_{{\omega},n}(z)=z^{2}\tilde{\psi}_{{\omega},n,d}(z)-\overline{W}_{{\omega},n,d}(z)$, where $\overline{W}_{{\omega},n,d}(z)$ is the reminder of $\ln W_{{\omega},n}(iz)$ when approximated by powers of $W_{{\omega},n,d}(iz)$. Note that $\varphi_{{\omega},n}(z)$ is analytic, that it vanishes at $z=0$ and that $\sup_{a\in B(0,r_{1})}|\varphi_{{\omega},n}(z)|\leq C_{1}$ ($P$-a.s.) for some positive constants $r_{1}$ and $C_{1}$ which do not depend on ${\omega}$. Set

[TABLE]

Since $\Pi_{{\omega},n,2}$ converges to ${\sigma}^{2}>0$ as $n\to\infty$, it follows from (4.5) that for any sufficiently small ${\delta}_{0}$, $P$-a.s. we have

[TABLE]

Combining this with (4.4), we derive that for any sufficiently small ${\delta}_{0}>0$ for $P$-a.a. ${\omega}$,

[TABLE]

Consider the functions $g_{{\omega},n}(t)=e^{-\frac{t^{2}\Pi_{{\omega},n,2}}{2}}(1+Q_{{\omega},n}(t))$. Using the fact that for any $c>0$, the function $\sqrt{2\pi c}(it)^{k}e^{-\frac{t^{2}c}{2}}$ is the Fourier transform of the function $e^{-\frac{t^{2}}{2c}}$, and using the integration by parts formula with integrals of the form $\int x^{j}e^{-cx^{2}}dx$, we derive that

[TABLE]

where F is the Fourier transform operator and all the $R_{{\omega},n,k}$’s are polynomials whose coefficients are linear combinations of the coefficients of the $A_{{\omega},n,k}$’s, and the degree of $R_{{\omega},n,k}$ depends only on $k$. Define $G_{{\omega},n}$ by G_{{\omega},n}(t)=\int_{-\infty}^{t}\textbf{F}\big{(}g_{{\omega},n}\big{)}(-\xi)d\xi. Then by the Fourier inversion theorem, $\textbf{F}(G^{\prime}_{{\omega},n})=g_{{\omega},n}$ and $\xi\to\sqrt{2\pi\Pi_{{\omega},n,2}}\textbf{F}(g_{{\omega},n})(-\xi)$ is the function inside the Fourier transform on the right hand side of (4.8). Using also the integration by parts formula with integrals of the form $\int t^{j}e^{-ct^{2}}dt=\int t^{j-1}te^{-ct^{2}}dt$, we derive that

[TABLE]

where all the $P_{{\omega},n,k}$’s are polynomials whose coefficients are linear combinations of the coefficients of the $A_{{\omega},n,k}$’s, and the degree of $P_{{\omega},n,k}$ depends only on $k$.

Next, applying (4.1) with $F=F_{{\omega},n}(s)=\mu_{\omega}\{x\in{\mathcal{E}}_{\omega}:S_{n}^{\omega}\leq\sqrt{n}s\}$ and $G=G_{{\omega},n}$, taking into account (4.7), we obtain that for any ${\varepsilon}>0$ and $B>\frac{A}{{\varepsilon}}$, where $A$ comes from (4.1),

[TABLE]

where

[TABLE]

We note that the polynomials $P_{{\omega},k,n}$ appearing in the definition of $G$ depend on $n$. Therefore, in principle, one cannot apply (4.1) directly. One can do it when $G^{\prime}_{{\omega},n}$ is uniformly bounded in $n$ and $\lim_{x\to\infty}\sup_{n\geq 1}|G_{{\omega},n}(x)|=0$ (these conditions are clearly satisfied in our circumstances). In (4.7) we have shown that $I_{1}=o(n^{-\frac{d}{2}})$. Since $\Pi_{{\omega},n,2}$ converges to ${\sigma}^{2}>0$ and the coefficients of all $A_{{\omega},k}$’s in the definition of $Q_{{\omega},n}$ are bounded in $n$, it follows that $I_{3}={\mathcal{O}}(e^{-cn})$ for some $c>0$. In order to complete the proof of Theorem 2.5, it is sufficient to show that $I_{2}=o(n^{-\frac{d}{2}})$. Recall that

[TABLE]

for any $t\in{\mathbb{R}}$. In the case when $d=1$, using Assumption 2.3 with $r=\frac{d}{2}$ and the compact set $J=\{t\in{\mathbb{R}}:{\delta}_{0}\leq|t|\leq B\}$ we obtain that

[TABLE]

for some $c>0$, which completes the proof of Theorem 2.5 (i). When $d>1$, in order to complete the proof of Theorem 2.5 (ii), it remains to show that, $P$-a.s. for any sufficiently large $C>0$

[TABLE]

This follows from the following lemma, together with the estimate $|\mu_{\omega}(e^{itS_{n}^{\omega}})|\leq\|{\mathcal{A}}_{it}^{{\omega},n}\|_{2}$, where $\|\cdot\|_{2}$ is the norm specified in Assumption 2.4.

4.1 Lemma**.**

Suppose that Assumption 2.4 holds true with $r=\frac{d}{2}$ and that $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic. Then for any $q>0$ there exist positive constants $K,D$ and $L$ so that $P$-a.s. for any sufficiently large $n$, and $t$ such that $K\leq|t|\leq Dn^{r}$,

[TABLE]

where $R_{\omega}$ is the random variable specified in Assumption 2.4.

Proof.

Set $r=\frac{d}{2}>0$ let $K_{\omega},D_{\omega},C_{\omega}$, $r_{2}({\omega})$, $R_{\omega}$ and $N({\omega})$ be the random variables specified in Assumption 2.4 with this specific $r$. Let $K,D,r_{2},C$ and $N$ be positive numbers so that the set

[TABLE]

has positive probability. Set $p_{0}=P({\Gamma})$. By Birkhoff’s ergodic theorem and Kac’s lemma applied with the transformation induced on $\Gamma$ by ${\theta}$, we obtain that $P$-a.s. there exists a sequence $m_{1}<m_{2}<m_{3}<...$ of positive integers so that

[TABLE]

It follows that for any $0<a<b<1$ and a sufficiently large $n$ there exists an index $k=k_{{\omega},n,a,b}$ so that $an<m_{k}<bn$. Indeed, let $1>{\varepsilon}>0$. Then for any sufficiently large $k$ we have $\frac{1-{\varepsilon}}{p_{0}}k<m_{k}<\frac{1+{\varepsilon}}{p_{0}}k$. Suppose that ${\varepsilon}$ is sufficiently small and let ${\delta}>0$ be so that $ap_{0}<{\delta}(1-{\varepsilon})<{\delta}(1+{\varepsilon})<bp_{0}$. Let $(k_{n})_{n=1}^{\infty}$ be a strictly increasing sequence of positive integers so that $k_{n}/n$ converges to ${\delta}$ as $n\to\infty$. Then for any sufficiently large $n$ we have

[TABLE]

Next, let $q>0$ and let $m$ be a positive integer such that $mr_{2}>r+q$. For any sufficiently large $n$ we can find $k_{1}(n),k_{2}(n),...,k_{m}(n)$ so that for each $i$,

[TABLE]

Set ${\omega}_{i}={\theta}^{m_{i}}{\omega}$, ${\Delta}_{i}(n)=k_{i+1}(n)-k_{i}(n)$, $i=1,2,...,m-2$, ${\Delta}_{0}(n)=k_{1}(n)-m_{1}$, ${\Delta}_{m-1}(n)=n-k_{m-1}(n)$ and $k_{0}(n)=1$. Then there exists a constant $c>0$ which depends only on $m$ so that

[TABLE]

Now, $P$-a.s for any large enough $n$ we can write for all $t$’s,

[TABLE]

If $K\leq|t|\leq Dn^{d}$ then, since ${\omega}_{i}\in{\Gamma}$ for all $i$’s, for any sufficiently large $n$ and $0\leq i\leq m-1$ we have

[TABLE]

By Assumption 2.4 we have $\|{\mathcal{A}}_{it}^{{\omega},m_{1}}\|_{2}\leq R_{\omega}(1+|t|)$ which is less than $R_{\omega}(1+Dn^{r})$ for $t$’s in the above range. We conclude from submultiplicity of norms of operators that $P$-a.s. for any sufficiently large $n$ and $t$ so that $K\leq|t|\leq Dn^{r}$ we have

[TABLE]

and the proof of the lemma is complete. ∎

Now we will explain how to derive Theorem 2.5 (iii). It follows from the definition of the $P_{{\omega},n,k}$’s that their coefficients are algebraic combinations of the derivatives at $z=0$ of the functions $\mu_{\omega}(h_{\omega}(z))$ and $n^{-1}\sum_{j=0}^{n-1}\Pi_{{\theta}^{j}{\omega}}(z)$. Since the derivatives of these functions [math] are uniformly bounded in ${\omega}$ and $n$ (when ${\omega}$ ranges over a set of full probability), these coefficient are uniformly bounded in ${\omega}$ and $n$, and the coefficients of $P_{{\theta}^{-n}{\omega},n,k}$ converge $P$-a.s. and in $L^{p}$ for any $p\in[1,\infty)$. When (3.9) holds true then all the coefficients of the polynomials $P_{{\theta}^{-n}{\omega},n,k},k=1,2,...$ converge towards their limits with rate of order $n^{-\frac{1}{2}}\ln n$.

5. Examples

5.1. Random distance expanding maps

Let $({\Omega},{\mathcal{F}},P,{\theta})$, $({\mathcal{X}},\rho)$ and ${\mathcal{E}}_{\omega}$ be as described at the beginning of Section 2, and let $\{T_{\omega}:{\mathcal{E}}_{\omega}\to{\mathcal{E}}_{{\theta}{\omega}},\,{\omega}\in{\Omega}\}$ be a collection of maps between the metric spaces ${\mathcal{E}}_{\omega}$ and ${\mathcal{E}}_{{\theta}{\omega}}$, so that the map $({\omega},x)\to T_{\omega}x$ is measurable with respect to the $\sigma$-algebra ${\mathcal{P}}$ which is the restriction of ${\mathcal{F}}\times{\mathcal{B}}$ on ${\mathcal{E}}$. Consider the skew product transformation $T:{\mathcal{E}}\to{\mathcal{E}}$ given by

[TABLE]

For any ${\omega}\in{\Omega}$ and $n\in{\mathbb{N}}$ consider the $n$-th step iterates $T_{\omega}^{n}$ given by

[TABLE]

Our additional requirements concerning the family of maps $\{T_{\omega}:{\omega}\in{\Omega}\}$ are collected in the following assumptions.

5.1 Assumption** (Topological exactness).**

There exist a constant $\xi>0$ and a random variable $n_{\omega}\in{\mathbb{N}}$ such that $P$-a.s.,

[TABLE]

where for any ${\omega}\in{\Omega}$, $x\in{\mathcal{E}}_{\omega}$ and $r>0$, $\,B_{\omega}(x,r)$ denotes a ball in ${\mathcal{E}}_{\omega}$ around $x$ with radius $r$.

5.2 Assumption** (The pairing property).**

There exist random variables ${\gamma}_{\omega}>1$ and $D_{\omega}\in{\mathbb{N}}$ such that $P$-a.s. for any $x,x^{\prime}\in{\mathcal{E}}_{{\theta}{\omega}}$ with $\rho(x,x^{\prime})<\xi$ we can write

[TABLE]

where $\xi$ is specified in Assumption 5.1,

[TABLE]

and

[TABLE]

for any $1\leq i\leq k$.

These assumptions hold true for the random distance expanding maps considered in [27] since in the setup there the inverse branches are contracting, see Section 2.5 in [27]. We refer the readers to Section 2.1 in [27] for a description of several examples of such maps $T_{\omega}$.

According to Lemma 4.11 in [27] (applied with $r=\xi$), there exist an integer valued random variable $L_{\omega}\geq 1$ and ${\mathcal{F}}$-measurable functions ${\omega}\to x_{{\omega},i}\in{\mathcal{X}},\,i=1,2,3,...$ so that $x_{{\omega},i}\in{\mathcal{E}}_{\omega}$ for each $i$ and

[TABLE]

Note that $L_{\omega}$ is constant in ${\omega}$ when ${\mathcal{E}}_{\omega}$ does not depend on ${\omega}$ (i.e. when ${\mathcal{E}}={\Omega}\times{\mathcal{Y}}$ for an appropriate ${\mathcal{Y}}\subset{\mathcal{X}}$).

Next, for any $g:{\mathcal{E}}\to{\mathbb{C}}$ and ${\omega}\in{\Omega}$ consider the function $g_{\omega}:{\mathcal{E}}_{\omega}\to{\mathbb{C}}$ given by $g_{\omega}(x)=g({\omega},x)$. Let $\phi,u:{\mathcal{E}}\to{\mathbb{R}}$ be measurable functions so that for $P$-a.a. ${\omega}$ the functions $\phi_{\omega}$ and $u_{\omega}$ are Hölder continuous with exponent ${\alpha}\in(0,1]$ which is independent of ${\omega}$. Let $z\in{\mathbb{C}}$ and consider the transfer operators ${\mathcal{L}}_{z}^{\omega},\,{\omega}\in{\Omega}$ which map functions on ${\mathcal{E}}_{\omega}$ to functions on ${\mathcal{E}}_{{\theta}{\omega}}$ by the formula

[TABLE]

Note that under Assumption 5.2 the operators ${\mathcal{L}}^{\omega}_{z},\,z\in{\mathbb{C}}$ are well defined and since $\phi_{\omega}$ and $u_{\omega}$ are Hölder continuous they map a continuous function on ${\mathcal{E}}_{\omega}$ to a continuous function on ${\mathcal{E}}_{{\theta}{\omega}}$. For any $n\in{\mathbb{N}}$ and $z\in{\mathbb{C}}$ consider the $n$-th step iterates ${\mathcal{L}}_{z}^{{\omega},n}$ of the transfer operator given by

[TABLE]

Then

[TABLE]

where $S_{n}^{\omega}\psi=\sum_{i=0}^{n-1}\psi_{{\theta}^{i}{\omega}}\circ T_{\omega}^{i}$ for any function $\psi:{\mathcal{E}}\to{\mathbb{C}}$, ${\omega}\in{\Omega}$ and $n\geq 1$. We also consider the (global) transfer operator ${\mathcal{L}}_{z}$ acting on functions $g:{\mathcal{E}}\to{\mathbb{C}}$ by the formula

[TABLE]

namely ${\mathcal{L}}_{z}$ is generated by the skew product map $T$ and the function $\phi+zu$. Next, let $g:{\mathcal{E}}\to{\mathbb{C}}$ be a measurable function. Let ${\alpha}$ as described before (5.7) and set

[TABLE]

where $\|\cdot\|_{\infty}$ is the supremum norm and \rho^{\alpha}(x,x^{\prime})=\big{(}\rho(x,x^{\prime})\big{)}^{\alpha}. These norms are ${\mathcal{F}}$-measurable as a consequence of Lemma 5.1.3 in [14]. We denote by ${\mathcal{H}}_{{\alpha},\xi}^{\omega}=H_{1}^{\omega}$ the space of all functions $f:{\mathcal{E}}_{\omega}\to{\mathbb{C}}$ so that $\|f\|_{{\alpha},\xi}<\infty$.

Our additional requirements from $T_{\omega},\phi_{\omega}$ and $u_{\omega}$ are specified in the following

5.3 Assumption**.**

(i) The random variables $n_{\omega},D_{\omega},L_{\omega}$ are bounded and ${\gamma}_{\omega}-1$ is bounded from below by some positive constant.

(ii) For any measurable function $g:{\mathcal{E}}\to{\mathbb{C}}$ so that $\|g_{\omega}\|_{{\alpha},\xi}\leq 1$ the map $({\omega},x)\to{\mathcal{L}}_{0}g({\omega},x)$ is measurable.

(iii) For $P$-a.a. ${\omega}$ we have $\phi_{\omega},u_{\omega}\in{\mathcal{H}}_{\omega}^{{\alpha},\xi}$ and the random variables $\|\phi_{\omega}\|_{{\alpha},\xi}$ and $\|u_{\omega}\|_{{\alpha},\xi}$ are bounded.

Assumption 5.3 (i) holds true, for instance when ${\mathcal{E}}_{\omega}={\mathcal{X}}=[0,1)$ (and then $L_{\omega}$ does not depend on ${\omega}$), and each $T_{\omega}$ is a piecewise distance expanding map on the unit interval with a bounded in ${\omega}$ number of monotonicity intervals (on which $T_{\omega}$ is onto $[0,1)$). Several multidimensional examples such as the case when $T_{\omega}x=(m_{1}({\omega})x_{1},...,m_{d}({\omega})x_{d})\,\text{mod }1$ can be also be given (here $m_{i}({\omega})\geq 2$ are integer valued and are bounded in ${\omega}$). We refer again the readers to Section 2.1 in [27] for more examples of this nature. Different type of examples are random topologically mixing subshifts of finite type with bounded number of symbols, see [22] for a description of such systems. Note that under Assumption 5.3 we have that ${\mathcal{L}}_{z}^{{\omega},n}({\mathcal{H}}_{\omega}^{{\alpha},\xi})\subset{\mathcal{H}}_{{\theta}^{n}{\omega}}^{{\alpha},\xi}$ and the corresponding operator norm satisfies $\|{\mathcal{L}}_{it}^{{\omega},n}\|_{{\alpha},\xi}\leq B(1+|t|)$ where $B$ is some constant (see Lemma 5.6.1 in [14]).

When the random variable $\|\phi_{\omega}\|_{{\alpha},\xi}$ is bounded then Assumption 5.3 (ii) holds true if for any set $A\subset{\mathcal{X}}$ with a sufficiently small diameter the set

[TABLE]

is measurable (note that when ${\mathcal{E}}_{\omega}={\mathcal{X}}$ then $\textbf{A}=T({\Omega}\times A)$ and that $T_{\omega}$ is injective on ${\mathcal{E}}_{\omega}\cap A$ if the diameter of $A$ is smaller than $\xi$). Indeed, in these circumstances we can approximate (uniformly in ${\omega}$) the function $g_{\omega}\cdot e^{\phi_{\omega}}$, by sums of the form $\sum_{i=1}^{m}a_{i}({\omega}){\mathbb{I}}_{A_{i}\cap{\mathcal{E}}_{\omega}}$, where $\{A_{i}\}$ is a partition of ${\mathcal{X}}$ into sets with small diameter, ${\mathbb{I}}_{A_{i}\cap{\mathcal{E}}_{\omega}}$ is the indicator set of $A_{i}\cap{\mathcal{E}}_{\omega}$ and $a_{i}({\omega})$ are some random variables whose absolute value is bounded by $1$. Then ${\mathcal{L}}_{0}$ can be approximated by

[TABLE]

which are measureable functions of $({\omega},x)$. Observe that it is enough to assume that the A’s are measurbale for sets $A$ which are generated by finite collections of open balls using only the basic Boolean operations. For such sets A is measurable if ${\Omega}$ is a topological space, ${\mathcal{F}}$ is the Borel ${\sigma}$-algebra, ${\mathcal{E}}_{\omega}={\mathcal{X}}$ and $T$ is continuous. Another example is the case when $T_{\omega}$ takes only a countable number of “values” $\{T_{1},T_{2},...\}$. We refer the readers to Chapter 4 in [27] for more aspects of such measurability issues.

5.1.1. Asymptotic moments: verification of Assumption 2.1

We recall first the following random complex RPF theorem, which was stated in [14] as Corollary 5.4.2.

5.4 Theorem**.**

Suppose that Assumptions 5.1, 5.2 and 5.3 hold true. Then there exists $\rho>0$ so that $P$-a.s. for any $z\in U:=\{\zeta\in{\mathbb{C}}:|\zeta|<\rho\}$ there is a unique triplet consisting of a nonzero complex number $\boldsymbol{{\lambda}}_{\omega}(z)$ a function $\textbf{h}_{\omega}(z)\in{\mathcal{H}}^{{\alpha},\xi}_{\omega}$ and a continuous linear functional $\boldsymbol{\nu}_{\omega}(z)$ on ${\mathcal{H}}^{{\alpha},\xi}_{\omega}$ so that $\boldsymbol{\nu}_{\omega}(z)\textbf{1}=1$,

[TABLE]

For any $z\in U$ the maps ${\omega}\to\boldsymbol{{\lambda}}(z)$ and $({\omega},x)\to\textbf{h}_{\omega}(z)(x),\,({\omega},x)\in{\mathcal{E}}$ are measurable and the family $\boldsymbol{\nu}_{\omega}(z)$ is measurable in ${\omega}$. When $z=t\in{\mathbb{R}}$ then $\boldsymbol{{\lambda}}_{\omega}(t)>0$, the function $\textbf{h}_{\omega}(t)$ is strictly positive, $\boldsymbol{\nu}_{\omega}(t)$ is a probability measure and the equality \boldsymbol{\nu}_{{\theta}{\omega}}(t)\big{(}{\mathcal{L}}_{t}^{\omega}g)=\boldsymbol{{\lambda}}_{\omega}(t)\boldsymbol{\nu}_{{\omega}}(t)(g) holds true for any bounded Borel function $g:{\mathcal{E}}_{\omega}\to{\mathbb{C}}$ and when the maps $T_{\omega}$ satisfy the assumptions from [27] then the triplet $(\boldsymbol{{\lambda}}_{\omega}(t),\textbf{h}_{\omega}(t),\boldsymbol{\nu}_{\omega}(t))$ coincides with the one constructed there with the function $\phi+tu$.

Moreover, this triplet is analytic and uniformly bounded around [math]. Namely, the maps

[TABLE]

are analytic, where $({\mathcal{H}}_{\omega}^{{\alpha},\xi})^{*}$ is the dual space of ${\mathcal{H}}_{\omega}^{{\alpha},\xi}$, and there is a constant $C_{0}>0$ so that

[TABLE]

where $\|\nu\|_{{\alpha},\xi}$ is the operator norm of a linear functional $\nu:{\mathcal{H}}_{\omega}^{{\alpha},\xi}\to{\mathbb{C}}$.

Furthermore, there exists constants $C>0$ and ${\delta}\in(0,1)$ so that $P$-a.s. for any $n\geq 1$ and $g\in{\mathcal{H}}^{{\alpha},\xi}_{\omega}$ we have

[TABLE]

Note that in [14] we have assumed that ${\mathcal{L}}_{0}g$ is measurable for any measurable function $g$, which was only needed in order to insure that the RPF triplets are measurable in ${\omega}$, but taking a careful look at the arguments in Chapter 4 of [14] we see that only Assumption 5.3 (ii) is needed.

We remark that the random (Gibbs) measure $\mu_{\omega}=\textbf{h}_{\omega}(0)\boldsymbol{\nu}_{\omega}(0)$ satisfies $(T_{\omega})_{*}\mu_{\omega}=\mu_{{\theta}{\omega}}$, or, equivalently, $\mu=\int_{{\Omega}}\mu_{\omega}dP({\omega})$ is $T$-invariant, where $T({\omega},x)=({\theta}{\omega},T_{\omega}x)$ is the skew product map. Consider now the transfer operator operator ${\mathcal{A}}_{z}^{\omega}$ generated by the map $T_{\omega}$ and the function $\tilde{\phi}_{\omega}+zu_{\omega}$, where $\tilde{\phi}_{\omega}=\phi_{\omega}+\ln\textbf{h}_{\omega}(0)-\ln\textbf{h}_{{\theta}{\omega}}(0)\circ T_{\omega}-\ln\boldsymbol{{\lambda}}_{{\omega}}(0)$. Then ${\mathcal{A}}_{z}^{\omega}\textbf{1}=\textbf{1}$,

[TABLE]

${\mathcal{A}}_{z}^{{\omega},n}({\mathcal{H}}_{\omega}^{{\alpha},\xi})\subset{\mathcal{H}}_{{\theta}^{n}{\omega}}^{{\alpha},\xi}$ and ${\mathcal{A}}_{z}^{\omega}$ also satisfies the above random complex RPF theorem. More precisely, Assumption 2.3 is satisfied with the norm $\|\cdot\|_{1}=\|\cdot\|_{{\alpha},\xi}$ and triplets $({\lambda}_{\omega}(z),h_{\omega}(z),\nu_{\omega}(z))$ which also satisfy ${\lambda}_{\omega}(0)=1$, $h_{\omega}(0)\equiv\textbf{1}$ and $\nu_{\omega}(0)=\mu_{\omega}$. We refer the readers to the begging of Section 4 in [17] for the precise details. Since $\|{\mathcal{A}}_{it}^{{\omega},n}\textbf{1}\|_{{\alpha},\xi}\leq C\|{\mathcal{L}}_{it}^{{\omega},n}\textbf{1}\|_{{\alpha},\xi}$ for some constant $C>0$ (see [14], Ch. 5), it is sufficient to show that ${\mathcal{L}}_{z}^{\omega}$ satisfy Assumption 2.3 with the above norm. Henceforth, we will always assume that $\mu_{\omega}(u_{\omega})=0$ for $P$-a.a. ${\omega}$, which, due to $T$-invariance of $\mu$, implies that $\mu_{\omega}(S_{n}^{\omega}u)=0$ for each $n\geq 1$. This is not really a restriction since we can always replace $u_{\omega}$ by $u_{\omega}-\int u_{\omega}(x)d\mu(x)$.

5.1.2. Edgeworth expansions of order one: verification of Assumption 2.3

The first part of Assumption 2.3 is satisfied due to the random Lasota-Yorke type inequality stated in Lemma 5.6.1 in [14]. Next, under the following assumption we showed in Chapter 7 of [14] that the rest of Assumption 2.3 holds true for the operators ${\mathcal{L}}_{z}^{\omega}$.

5.5 Assumption**.**

(i) ${\Omega}$ is a topological space, ${\mathcal{F}}$ is the corresponding Borel ${\sigma}$-algebra, $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic and ${\theta}$ has a periodic point, namely there exist ${\omega}_{0}\in{\Omega}$ and $n_{0}\in{\mathbb{N}}$ so that ${\theta}^{n_{0}}{\omega}_{0}={\omega}_{0}$. Moreover, $P(U)>0$ for any open set $U$ which intersects the finite orbit $\{{\theta}^{i}{\omega}_{0}:\,0\leq i<n_{0}\}$ and the spaces ${\mathcal{E}}_{\omega}$ are locally independent of ${\omega}$ around the points ${\theta}^{j}{\omega}_{0},\,0\leq j<n_{0}$.

(ii) The map ${\theta}$ is continuous at the points ${\omega}_{0},{\theta}{\omega}_{0},...,{\theta}^{n_{0}-1}{\omega}_{0}$, where ${\omega}_{0}$ and $n_{0}$ come from (i), and for any compact set $J\subset{\mathbb{R}}\setminus\{0\}$ the family of maps ${\omega}\to{\mathcal{L}}_{it}^{\omega},\,t\in J$ is equicontinuous continuous (with respect to the operator norm $\|\cdot\|_{{\alpha},\xi}$) at the points ${\theta}^{j}{\omega}_{0},\,0\leq j<n_{0}$.

(iii) The function $S_{n_{0}}^{{\omega}_{0}}u$ is non-arithmetic (aperiodic) with respect to the map $T^{n_{0}}_{{\omega}_{0}}$ in the classical sense of [20], namely if for any $t\in{\mathbb{R}}\setminus\{0\}$ there exist no nonzero $g\in{\mathcal{H}}_{{\omega}_{0}}^{{\alpha},\xi}$ and ${\lambda}\in{\mathbb{C}}$, $|{\lambda}|=1$ such that

[TABLE]

Note that under Assumption 5.5 we showed in Chapter 7 of [14] that, in fact, for any compact set $J\subset{\mathbb{R}}\setminus\{0\}$, $P$-.a.s for any $n\geq 1$ we have

[TABLE]

for some random variable $d_{\omega}>0$ and a constant $c>0$. In Chapter 7 of [14] we stated that (5.15) holds true under the additional assumption that $P(U)>0$ for any open set $U$, but in the proof we relied on that only for $U$’s which intersect the orbit $\{{\omega}_{0},{\theta}{\omega}_{0},...,{\theta}^{n_{0}-1}{\omega}_{0}\}$. The condition about continuity of ${\omega}\to{\mathcal{L}}_{it}^{\omega}$ holds true, for instance, when the maps ${\omega}\to\phi_{\omega},u_{\omega}\in{\mathcal{H}}^{{\alpha},\xi}_{\omega}$ are continuous at the points ${\theta}^{j}{\omega}_{0},\,0\leq j<n_{0}$ and the branches of $T_{\omega}$ are either locally constant as functions of ${\omega}$ there, or they are continuous at these points with respect to an appropriate topology. For instance, we can just assume that $T_{\omega}$ does not depend on ${\omega}$ around the points ${\theta}^{i}{\omega}_{0}$. This is the case when $({\Omega},{\mathcal{F}},{\sigma})$ is a shift system built over a countable alphabet $\mathscr{A}$ so that $T_{\omega}=T_{{\omega}_{0}}$ depends only on the [math]-th coordinate of ${\omega}=({\omega}_{i})$. We can consider any sub-shift of ${\Omega}$ instead, assuming it contains a point of the form $(...a,a,a,...)$ where $a\in\mathscr{A}^{n_{0}}$ for some $n_{0}$. Of course, we we also have to assume that $P(U)>0$ for the appropriate $U$’s. When considering interval maps $T_{\omega}$ with deterministic monotonicity intervals $I_{1},...,I_{D}$ (or similar multidimensional examples), then the condition about the continuity holds true if the restriction of the inverse of each map $T_{\omega}|_{I_{i}}$ is continuous in ${\omega}$ at ${\omega}={\theta}^{j}{\omega}_{0}$, $i=0,...,n_{0}-1$.

Note that when ${\mathcal{X}}$ is a smooth connected Riemannian manifold then we can also consider the norms $\|g\|_{C^{1}}=\sup|g|+\sup|Dg|$ instead of $\|\cdot\|_{{\alpha},\xi}$. In this case when each $T_{\omega}$ is also smooth and the map ${\omega}\to T_{\omega}$ is continuous in the $C^{1}$ topology then the transfer operators ${\omega}\to{\mathcal{L}}_{\omega}$ are continuous in the operator-norm topology (corresponding to the norm $\|\cdot\|_{C^{1}}$), see Proposition 5.3 in [9]. Therefore, when considering smooth functions $\phi_{\omega}$ and $u_{\omega}$ which are continuous in the $C^{1}$-topology as functions of ${\omega}$ we get that the conditions in item (ii) above are satisfied with $\|\cdot\|_{C^{1}}$ instead of $\|\cdot\|_{{\alpha},\xi}$, and all our results hold true.

5.1.3. Edgeworth expansions of high orders: verification of Assumption 2.4

Now we will explain in which circumstances Assumption 2.4 is satisfied with any $r$. We first recall the following results from [4]. Let $\mathscr{L}_{b},\,b\in{\mathbb{R}}$ be the deteministic transfer operators which are generated by two piecewise smooth expanding functions $f:{\mathbb{T}}\to{\mathbb{T}}$ and $\tau:{\mathbb{T}}\to{\mathbb{T}}$ and is given by the formula

[TABLE]

Let $\|\cdot\|_{2}=\|\cdot\|_{{\textbf{BV}}}=\|\cdot\|_{L^{1}({\mathbb{T}})}+\text{Var}(\cdot)$ be the norm in the space of functions with bounded variation. When $\tau$ is not cohomologous to a piecewise constant function (with respect to the base map $f$, see [4]), then by Proposition 1 in [4], there exist positive constants $b_{0},\rho$ and ${\gamma}_{2}$ such that

[TABLE]

where $\|h\|_{(b)}=(1+|b|)^{-1}\|h\|_{\textbf{BV}}+\int_{{\mathbb{T}}}|h(x)|dx$. In [25] the authors showed that this implies that Assumption 2.4 holds true in this deterministic case.

Next, we assume that ${\mathcal{E}}_{\omega}={\mathbb{T}}=[0,1)$ for any ${\omega}$ and that, in addition to the requirements at the beginning of Section 5.1, the functions $T_{\omega}$ and $u_{\omega}$ are piecewise smooth expanding functions and that $u_{\omega}$ takes values in ${\mathbb{T}}$. Then Assumptions 2.1 holds true. In order to keep our notations in line with the ones in [4], we set $f_{\omega}=T_{\omega}$ and $\tau_{\omega}=u_{\omega}$. We also set $\phi_{\omega}=-\ln|f_{\omega}^{\prime}|$. We assume here that there exists no random family $\eta_{\omega}$ of functions $\eta_{\omega}:{\mathbb{T}}\to{\mathbb{T}}$ so that $\tau_{\omega}-\eta_{{\theta}{\omega}}\circ f_{\omega}+\eta_{\omega}$ is piecewise constant ($P$-a.s.). Then, when $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic, an appropriate quenched version of Proposition 1 in [4] holds true when we replace $f^{n}$ and $\sum_{j=0}^{n-1}\tau\circ f^{j}$ with $f_{\omega}^{n}:=f_{{\theta}^{n-1}{\omega}}\circ f_{{\theta}^{n-2}{\omega}}\circ\cdots\circ f_{\omega}$ and $\sum_{j=0}^{n-1}\tau_{{\theta}^{j}{\omega}}\circ f_{\omega}^{j}$, respectively, when all the conditions in [4] hold true uniformly in ${\omega}$ in the following sense: there exist random variables $\rho_{\omega},{\gamma}_{2}({\omega}),b_{\omega}>0$ so that $P$-a.s., for any $b\in{\mathbb{R}}$,

[TABLE]

Before explaining how the proof of (5.17) differs from the proof of (5.16), we will rely on (5.17) in order to show that Assumption 2.4 holds true. Indeed, let $b_{0},\rho_{1},\rho_{2},{\gamma}_{2}>0$ be so that

[TABLE]

Then by ergodicity of $({\Omega},{\mathcal{F}},P,{\theta})$, for $P$-almost any ${\omega}$ there exists a sequence of positive integers $n_{1}<n_{2}<n_{3}<...$ so that for each $i$,

[TABLE]

and $\lim_{k\to\infty}k^{-1}n_{k}=a^{-1}$. It follows that for any $c>0$ there exists a constant $c_{1}>0$ so that for any sufficiently large $n$, there are indexes $i_{1}<i_{2}<...<i_{s}$, $s=s_{n}=[c_{1}\frac{n}{\ln n}]$, $i_{j}=i_{j}(n)$ with the property that $n_{i_{j}}+c\ln n<n_{i_{j+1}}<n-c\ln n$ for any $j=1,2,...,s-1$. In fact, since $n^{-1}\{\max k:n_{k}\leq n\}$ converges as $n\to\infty$ to $a$, we can always take $i_{1}=1$, $i_{s}<\frac{n}{2}-c\ln n$ and $c_{1}=\frac{a}{4(c+1)}$.

Next, it follows exactly as in Lemma 2 in [4] that $\|{\mathcal{L}}_{ib}^{{\omega},n}\|_{(b)}\leq C$ for any $b$ and $n$, where $C\geq 1$ is some constant (in fact, it is possible to take $C=1$, but it does not matter for our purpose). Let $d>0$ and let $b_{1}>\max(b_{0},1)$ be so that $b_{1}^{\frac{1}{2}\rho_{1}{\gamma}_{2}}>2C$. Then for any real $b$ so that $b_{1}\leq|b|\leq n^{d}$ we have $\rho_{2}\ln|b|\leq\rho_{2}d\ln:=c\ln n$. Let $c_{1}$ and $i_{1}<i_{2}<...<i_{s}$ (for a sufficiently large $n$) be as describe in the previous paragraph with $c=\rho_{2}d$. For $n$ large enough, set

[TABLE]

By writing

[TABLE]

and using submultiplicativity of norms of operators, we deduce that $P$-a.s. for any sufficiently large $n$ and real $b$ so that $b_{1}\leq|b|\leq n^{d}$ we have

[TABLE]

which implies that (2.2) from Assumption 2.4 holds true with $d_{1}=d$ and $K_{\omega}=b_{1}$ (since $|b|\leq n^{d}\leq 2^{2d\ln n}$).

Now we will explain the differences between the derivations of (5.16) and (5.17), and show that a random variable $R_{\omega}$ with the properties required in Assumption 2.4 exists. First, in the above circumstances by Theorem 2.2 in [23] we have $\boldsymbol{{\lambda}}_{\omega}(0)=1$ and $\boldsymbol{\nu}_{\omega}(0)=\text{Lebesgue}$. By taking ${\alpha}=1$, the arguments in Chapter 5 of [14] show that $\|h_{\omega}(0)\|_{\textbf{BV}}$ and $\|1/h_{\omega}(0)\|_{\textbf{BV}}$ are both bounded random variables. Therefore, in order to show that Assumption 2.4 hold true with the operators ${\mathcal{A}}_{it}^{{\omega},n}$, it is sufficient to show that it holds true with the operators ${\mathcal{L}}_{it}^{{\omega},n}$ (and the norm $\|\cdot\|_{2}=\|\cdot\|_{\textbf{BV}}$). Let $\Lambda$ and $\tilde{\lambda}>1$ be positive constants so that $P$-a.s.

[TABLE]

Then all the arguments in the proofs in Section 2 of [4] proceed similarly with the transfer operators ${\mathcal{L}}_{ib}^{{\omega},n}$ in place of $\mathscr{L}_{ib}^{n}$, since they only rely on the expansion properties of $f$ and $\tau$. In particular, $\|{\mathcal{L}}_{it}^{{\omega},n}\|_{\textbf{BV}}\leq R(1+|t|)$ for some constant $R>0$. Moreover, it is possible to construct partitions ${\Omega}_{n}={\Omega}_{n}^{\omega},\,n\geq 0$ almost exactly as in [4] (see the paragraph preceding Lemma 4 in [4]), which in our case will also depend on ${\omega}$: their recursive construction in our nondeterministic setup proceeds exactly as in [4], expect that in the $n$-th step of the construction we consider the function $f_{{\theta}^{n}{\omega}}$ in place of $f$. The estimates derived in Lemma 4 in [4] from there hold true also for the ${\Omega}_{n}^{\omega}$’s, and the proof goes exactly in the same way as in [4].

Now we will explain how to generalize the ideas from Section 3 in [4] to our non-deterministic setting. Set $F_{\omega}(x,u)=(f_{\omega}(x),u+\tau_{\omega}(x))$, where $x,y\in{\mathbb{T}}$. Then

[TABLE]

and the differential $DF_{\omega}(x,u)$ of $F_{\omega}$ does not depend on $u$. The fact that $f_{\omega}$ and $\tau_{\omega}$ are expanding uniformly in ${\omega}$ implies the cones ${\mathcal{K}}=\{({\alpha},{\beta}):|\beta||{\alpha}|^{-1}\leq C_{1}\}$ are invariant under each $DF_{\omega}$. Here $C_{1}$ is a constant so that $P$-a.s.,

[TABLE]

Thus, the notation of transversality from Section 3 in [4] is naturally extended to our setup, using $DF^{n}_{\omega}(x)$ in place of the differential of F^{n}(x,u)=\big{(}f^{n}(x),u+\sum_{j=0}^{n-1}\tau\circ f^{j}(x)\big{)}. That is, for any $y\in{\mathbb{T}}$, we will say that $x_{1},x_{2}\in(f_{\omega}^{n})^{-1}\{y\}$ are transversal if $DF^{n}_{\omega}(x_{1}){\mathcal{K}}\bigcap DF^{n}_{\omega}(x_{2}){\mathcal{K}}=\{0\}$. Set $J_{{\omega},n}=|(f_{\omega}^{n})^{\prime}|^{-1}$, and define the quantity $\varphi_{\omega}(n)$ by

[TABLE]

where $A_{\omega}(x_{1},n,y)$ is the set of all preimages of $y$ by $f_{\omega}^{n}$ which are not transversal to $x_{1}$. We also define the quantity $\tilde{\varphi}_{\omega}(n)$ by

[TABLE]

where $L\subset{\mathbb{R}}^{2}$ is a line which passes through the origin and $\textbf{h}_{\omega}=\textbf{h}_{\omega}(0)$ is the random function from the RPF triplet, i.e. the one satsifying that ${\mathcal{L}}_{0}^{\omega}\textbf{h}_{\omega}=\textbf{h}_{{\theta}{\omega}}$ (recall that $\boldsymbol{{\lambda}}_{\omega}(0)=1$). Note that (see Chapter 5 in [14]) the function $\textbf{h}_{\omega}$ is strictly positive and is bounded and bounded away from [math] uniformly in ${\omega}$. Then $\tilde{\varphi}_{\omega}(n)\leq 1$ and it satisfies that

[TABLE]

By Kingman’s subadditive ergodic theorem, the limit $\lim_{n\to\infty}(\tilde{\varphi}_{\omega}(n))^{\frac{1}{n}}$ exists, $P$-a.s., and it does not depend on ${\omega}$. When this limit equals $1$ then, by considering the subadditive sequence $g_{n}=\int\ln\tilde{\varphi}_{\omega}(n)dP({\omega})$, it follows that $\tilde{\varphi}_{\omega}(n)=1$ for $P$-a.a. ${\omega}$ and all $n$’s. Relying on that, the arguments from the proof of Lemma 7 in [4] show that

[TABLE]

if there exists no random family $\eta_{\omega}$ of functions $\eta_{\omega}:{\mathbb{T}}\to{\mathbb{T}}$ so that $\tau_{\omega}-\eta_{{\theta}{\omega}}\circ f_{\omega}+\eta_{\omega}$ is piecewise constant ($P$-a.s.). The rest of the derivation of (5.17) proceeds now as in Section 4 of [4], relying on the uniform (in ${\omega}$) expansion properties of the functions $f_{\omega}$ and $\tau_{\omega}$.

5.6 Remark**.**

The assumption that $u_{\omega}$ is continuous can be dropped since all the results from [14] hold true also when $u_{\omega}$ is piecewise Hölder continuous. We will not present here all the details and instead refer the readers to Assumption 5.2.1 in [14], and to the results proceeding it.

5.1.4. Convergence rates towards the asymptotic moments

In the circumstances of Section 3.1, the operators ${\mathcal{A}}_{z}^{{\omega}}$ do not depend only on the [math]-th coordinate even when ${\mathcal{L}}_{z}^{\omega}$ depends on it (only). Still, in order to obtain the convergence rates stated in Section 2.1, it is enough to obtain the appropriate almost sure rates in the convergence of the ergodic averages $n^{-1}\sum_{j=0}\Pi_{{\theta}^{j}{\omega}}^{(k)}(0)$ (see the arguments in Section 3. Let us denote by $\boldsymbol{\Pi}_{\omega}(z)$ the pressure function corresponding to $\boldsymbol{{\lambda}}_{{\omega}}(z)$. In (4.2) in [17] we have proved that there exist constants $c_{1}$ and $\rho_{1}$ so that $P$-a.s. for any $z\in{\mathbb{C}}$ with $|z|<\rho_{1}$ and any $n\geq 1$ we have

[TABLE]

Note that it is indeed possible to bound from below $\min|h_{\omega}(z)|$ when $z$ belongs to some deterministic neighborhood of the origin since $h_{\omega}(x)\geq C>0$ for some constant $C>0$ which does not depend on ${\omega}$ and $x$ (see (5.9.4) and (4.3.32) in [14]). Therefore, when ${\mathcal{L}}_{z}^{\omega}$ depend only on the [math]-th coordinate, we can just repeat the argument from Section 3.1 with $\boldsymbol{\Pi}$ in place of $\Pi$ we obtain all the results stated in Section 2.1 under the mixing conditions from Section 3.1.

5.2. Markov chains with transition densities

For any ${\omega}\in{\Omega}$ denote by $B_{\omega}=H_{1}^{\omega}$ the Banach space of all bounded Borel functions $g:{\mathcal{E}}_{\omega}\to{\mathbb{C}}$ together with the supremum norm $\|\cdot\|_{\infty}$. For any $g:{\mathcal{E}}\to{\mathbb{C}}$ consider the functions $g_{\omega}:{\mathcal{E}}_{\omega}\to{\mathbb{C}}$ given by $g_{\omega}(x)=g({\omega},x)$. Then by Lemma 5.1.3 in [14], the norm ${\omega}\to\|g_{\omega}\|_{\infty}$ is a ${\mathcal{F}}$-measurable function of ${\omega}$, for any measurable $g:{\mathcal{E}}\to{\mathbb{C}}$.

Let $r_{\omega}=r_{\omega}(x,y):{\mathcal{E}}_{\omega}\times{\mathcal{E}}_{{\theta}{\omega}}\to[0,\infty),\,{\omega}\in{\Omega}$ be a family of integrable in $y$ Borel measurable functions, $m_{\omega},\,{\omega}\in{\Omega}$ be a family of Borel probability measures on ${\mathcal{E}}_{\omega}$ and $u:{\mathcal{E}}\to{\mathbb{R}}$ be a measurable function so that $u_{\omega}\in B_{\omega},\,$ $P$-a.s. and that the random variable $\sup|u_{\omega}|=\|u_{\omega}\|_{\infty}$ is bounded. Consider the family of random operators $R_{z}^{\omega},\,z\in{\mathbb{C}}$ which map (bounded) Borel functions $g$ on ${\mathcal{E}}_{{\theta}{\omega}}$ to Borel measurable functions on ${\mathcal{E}}_{\omega}$ by the formula

[TABLE]

We will assume that $R_{0}^{\omega}$ are Markov operators, namely that $R_{0}^{\omega}\textbf{1}=\textbf{1}$ where 1 is the function which takes the constant value $1$ on ${\mathcal{E}}_{{\theta}{\omega}}$. Observe that

[TABLE]

and therefore for $P$-a.a. ${\omega}$ we have $\|R_{z}^{\omega}\|_{\infty}<\infty$ for any $z\in{\mathbb{C}}$, namely, $R_{z}^{\omega}$ is a continuous linear operator between the Banach spaces $B_{{\theta}{\omega}}$ and $B_{\omega}$.

5.7 Assumption**.**

The functions $r_{\omega}$ are measurable in ${\omega}$ and the the measures $m_{\omega}$ are measurable in the sense that the map ${\omega}\to\int_{{\mathcal{E}}_{{\omega}}}g_{{\omega}}(x)dm_{{\theta}{\omega}}(x)$ is measurable for any measurable function $g:{\mathcal{E}}\to{\mathbb{C}}$ so that $g_{\omega}(\cdot)$ is bounded $P$-a.s.

Under this assumption the map $({\omega},x)\to R_{0}^{\omega}g_{{\theta}{\omega}}(x)$ is measurable for any measurable function $g=g({\omega},x)=g_{\omega}(x)$.

For any ${\omega}\in{\Omega}$, $n\in{\mathbb{N}}$ and $z\in{\mathbb{C}}$ consider the $n$-th order iterates $R_{z}^{{\omega},n}:B_{{\theta}^{n}{\omega}}\to B_{\omega}$ given by

[TABLE]

Then we can write

[TABLE]

for some family $r_{\omega}(n,\cdot,\cdot)=r_{\omega}(n,x,y):{\mathcal{E}}_{\omega}\times{\mathcal{E}}_{{\theta}^{n}{\omega}}\to[0,\infty)$ of integrable in $y$ Borel measurable functions. We will assume that the following random version of the two sided Doeblin condition holds true.

5.8 Assumption**.**

There exist a bounded random variable $j_{\omega}\in{\mathbb{N}}$ and random variables $0<{\alpha}_{m}({\omega})\leq 1$, $m\in{\mathbb{N}}$ such that $P$-a.s.,

[TABLE]

for any $m\geq j_{\omega}\,$, $x\in{\mathcal{E}}_{\omega}$ and $y\in{\mathcal{E}}_{{\theta}^{m}{\omega}}$. Moreover, let $j_{0}$ be so that $j_{\omega}\leq j_{0}$, $P$-a.s. Then there exists ${\alpha}>0$ so that ${\alpha}_{j_{0}}({\omega})\geq{\alpha}$.

Under the above assumptions, we showed in Chapter 6 of [14] that the family of operators ${\mathcal{A}}_{z}^{\omega}=R_{z}^{{\theta}^{-1}{\omega}}$ satisfies Assumption 2.1 with the measure preserving system $({\Omega},{\mathcal{F}},P,{\theta}^{-1})$.

Let $\mu_{\omega}$ be any probability measure on ${\mathcal{E}}_{\omega}$ and let $\xi_{n}^{{\theta}^{n}{\omega}},\,n\geq 1$ be the Markov chain with initial distribution $\mu_{\omega}$ whose $n$-th step operator is given by $R_{0}^{{\omega},n}$. Set

[TABLE]

Then,

[TABLE]

and therefore all the results stated in Section 2 hold true with the random variables $S_{n}^{{\theta}^{-n}{\omega}}$.

5.2.1. Edgeworth expansions of order one: verification of Assumptions 2.3

The first part of Assumption 2.3 holds true with the supremum norm since $\|R_{it}^{\omega}\|_{\infty}\leq\|R_{0}^{\omega}\|_{\infty}=1$. Let ${\omega}_{0}\in{\Omega}$ and $n_{0}\in{\mathbb{N}}$ be so that ${\theta}^{n_{0}}{\omega}_{0}={\omega}_{0}$. Suppose that for any $t\in{\mathbb{R}}\setminus\{0\}$ the spectral radius $r(t)$ of the operator $R_{it}^{{\omega}_{0},n_{0}}$ is strictly less than $1$. We refer the reader to Section 6.3 in [25] for conditions guaranteeing that $r(t)<1,t\not=0$ (see the verification of Assumption (A3) there). Then by Lemma 2.10.4 in [14] the second part Assumption 2.3 holds true when Assumption 5.5 is satisfied with $R_{z}^{\omega}$ in place of ${\mathcal{L}}_{z}^{\omega}$ and with the norm $\|\cdot\|_{\infty}$ instead of $\|\cdot\|_{{\alpha},\xi}$. Note that in Chapter 7 we have also verified Assumption 2.3 under certain type of non-lattice condition. These results were obtained under the assumption that $P(j_{\omega}=j_{{\theta}{\omega}}=1)>0$ (see Assumption 7.1.4 there), and it is also possible to extend them to more general cases (see Section 7.4.4 there).

5.2.2. Edgeworth expansions of high orders: verification of Assumptions 2.4

We will show that Assumption 2.4 holds true with $H_{2}^{\omega}=B_{\omega}$ under the following

5.9 Assumption**.**

The system $({\Omega},{\mathcal{F}},P,{\theta})$ is ergodic and the probability that

[TABLE]

is positive.

Before verifying Assumption 2.4 we will provide conditions guaranteeing it holds true. The following result is proved exactly as in [25].

5.10 Lemma**.**

Let ${\omega}\in{\Omega}$ and suppose that ${\mathcal{E}}_{\omega}={\mathcal{E}}_{{\theta}{\omega}}$ is a compact connected smooth manifold, that $u_{{\theta}{\omega}}$ and $u_{{\theta}^{2}{\omega}}$ are piecewise smooth and that $r_{\omega}$ is a $C^{1}$-function. Moreover, assume that $j_{\omega}=j_{{\theta}{\omega}}=1$ and that the function $(y_{1},y_{2})\to u_{{\theta}{\omega}}(y_{1})+u_{{\theta}^{2}{\omega}}(y_{2})$ is not piecewise constant. Then condition (5.21) holds true with the above ${\omega}$.

Assumption 5.21 holds true when the probability that ${\omega}$ satisfies the conditions of Lemma 5.10 is positive.

Now we will verify Assumption 2.4. Under Assumption 5.9, there exist ${\varepsilon}_{0},r_{0}>0$ so that $p_{0}=P({\Gamma}_{0})>0$, where ${\Gamma}_{0}$ is the set of all ${\omega}$’s so that

[TABLE]

Since ${\theta}$ is ergodic $P$-a.s. there exists a sequence $n_{1}<n_{2}<n_{3}<...$ of positive integers so that

[TABLE]

Set $m_{i}=n_{2i}$, ${\omega}_{i}={\theta}^{-m_{i}}{\omega}$ and write

[TABLE]

where $k_{n}=\max\{i:m_{i}\leq n-3\}$. Since $\|R_{it}^{\omega}\|_{\infty}\leq 1$ and $k_{n}/n$ converges to $\frac{1}{2}p_{0}$ as $n\to\infty$, we obtain from submultiplicativity of norms of operators that $P$-a.s., for any $t\in{\mathbb{R}}$ so that $|t|\geq r_{0}$ and any sufficiently large $n$,

[TABLE]

which implies that Assumption 2.4 holds true. We note that Assumption 2.4 also holds true with the norms $\|\cdot\|_{\infty}$ when appropriate random versions of conditions $(C)$ and $(B_{k})$ from [29] are satisfied (with positive probability).