1 Introduction and a preview of the main results
1.1 Quenched limit theorems for random dynamical systems
Let $(X,{\mathcal B},m)$ be a probability space and let $({\Omega },{\mathcal F},{\mathbb P},{\sigma })$ be an invertible ergodic probabilitypreserving system. Let $T_{\omega }:X\to X,\,{\omega }\in {\Omega }$ , be a family of nonsingular maps (that is, ${m\circ T_\omega ^{1}\ll m}$ ) so that the corresponding skew product $\tau $ given by $\tau ({\omega },x)=({\sigma }{\omega },T_{\omega } x)$ is measurable. A random dynamical system is formed by the sequence of compositions
taken along the orbit of a ‘random’ point ${\omega }$ . The system $({\Omega },{\mathcal F},{\mathbb P},{\sigma })$ is often referred to as the driving system, and the map ${\sigma }$ is often referred to as the base map.
Let $\varphi :{\Omega }\times X\to {\mathbb R}$ be a measurable function (an ‘observable’) and let $\mu $ be a $\tau $ invariant probability measure on ${\Omega }\times X$ . Then $\mu $ can be decomposed as ${\mu =\int \mu _{\omega } \,d{\mathbb P}({\omega })}$ , where $\mu _{\omega }$ is a family of probability measures on X so that ${(T_{\omega })_*\mu _{\omega }=\mu _{{\sigma }{\omega }}}$ for ${\mathbb P}$ almost every (a.e.) ${\omega }$ . Set $S_n\varphi =\sum _{j=0}^{n1}\varphi \circ \tau ^j$ . Then
where $\varphi _{\omega }(\cdot )=\varphi ({\omega },\cdot )$ . For ${\mathbb P}$ a.e. ${\omega }$ we can consider the sequence of functions $S_n^{\omega }\varphi (\cdot )$ on the probability space $(X,{\mathcal B},\mu _{\omega })$ as random variables. Limit theorems for such sequences are called quenched limit theorems. Among the first papers dealing with quenched limit theorems for random dynamical systems are [Reference Kifer36, Reference Kifer37], where in [Reference Kifer36] a quenched largedeviations principle was obtained, and in [Reference Kifer37] a central limit theorem (CLT) and a law of the iterated logarithm were established. Since then quenched limit theorems for random dynamical systems have been extensively studied. For instance, in [Reference Dragičević, Froyland, GonzálezTokman and Vaienti16, Reference Dragičević, Hafouta, Pollicott and Vaienti20–Reference Dragičević, Hafouta and Sedro22] almost sure invariance principle (ASIP, an almost sure approximation by a sum of independent Gaussians) was established for random expanding or hyperbolic maps $T_{\omega }$ , in [Reference Dragičević and Hafouta19, Reference Hafouta and Kifer31] Berry–Esseen theorems (optimal rates in the CLT) were obtained for similar classes of maps and in [Reference Dragičević, Froyland, GonzálezTokman and Vaienti17, Reference Dragičević, Froyland, GonzálezTokman and Vaienti18, Reference Dragičević and Sedro23, Reference Hafouta and Kifer31] local CLTs were achieved. In addition, in [Reference Hafouta27] several limit theorems were extended to random nonuniformly hyperbolic or expanding maps. We would also like to refer to [Reference Alves, Bahsoun and Ruziboev3] for related results concerning mixing rates for random nonuniformly hyperbolic maps and to [Reference Haydn, Nicol, Török and Vaienti32] for related results concerning sequential dynamical systems, where an ASIP was obtained. We note that in many of the examples these results are obtained for the unique measure $\mu $ such that $\mu _{\omega }$ is absolutely continuous with respect to m. However, some results hold true even for maps $T_{\omega }:{\mathcal E}_{\omega }\to {\mathcal E}_{\sigma {\omega }}\subset X$ which are defined on random subsets of X (see [Reference Kifer, Lui, Hasselblatt and Katok40]), where in this case the most notable choices of $\mu _{\omega }$ are the socalled random Gibbs measures (see [Reference Hafouta and Kifer31, Reference Mayer, Skorulski and Urbański44]).
1.2 Limit theorem skew products
Let us consider the sums $S_n\varphi =\sum _{j=0}^{n1}\varphi \circ \tau ^j$ as random variables on the probability space $({\Omega }\times X,{\mathcal F}\times {\mathcal B},\mu )$ . In this paper will focus on limit theorems for such sequences of random variables. In order to demonstrate the difference between such limit theorems and the quenched ones, let us focus of the CLT. The quenched CLT means that for ${\mathbb P}$ a.e. ${\omega }$ , for all real t, we have
where ${\sigma }\geq 0$ is the number that satisfies ${\sigma }^2=\lim _{n\to \infty }(1/n)\mathrm {Var}_{\mu _{\omega }}(S_n^{\omega }\varphi )$ for ${\mathbb P}$ a.e. ${\omega }$ (assuming that this limit exists and does not depend on ${\omega }$ , refer to [Reference Kifer37, Theorem 2.3] for sufficient conditions). On the other hand, the CLT for the skew product means that for all real t we have
where ${\Sigma }^2=\lim _{n\to \infty }(1/n)\mathrm {Var}_{\mu }(S_n\varphi )$ . Note that, in contrast to the quenched case, the summands $X_j=\varphi \circ \tau ^j$ form a stationary sequence and, in applications, the existence of the limit ${\Sigma }^2$ follows from a sufficiently fast decay of $\mathrm {Cov}(X_0,X_n)$ as $n\to \infty $ . We also remark that both CLT’s above are formulated when ${\sigma }$ and ${\Sigma }$ are positive, and when one of them vanishes the convergence is towards the constant function $0$ .
When $\mu _{\omega }(\varphi _{\omega })$ does not depend on ${\omega }$ , we have that $\mu _{\omega }(\varphi _{\omega })=\mu (\varphi )$ and ${\sigma }^2={\Sigma }^2$ . In this case the quenched CLT implies the CLT for $S_n\varphi $ by integrating $\mu _{\omega }(\{x: S_n^{\omega }\varphi (x)\mu _{\omega }(S_n^{\omega }\varphi )\leq t\sqrt n\})$ with respect to ${\mathbb P}$ (and similarly other distributive limit theorems for the skew product follow from the quenched ones). However, it is less likely to be true when $\mu _{\omega }(\varphi _{\omega })$ depends on ${\omega }$ . Remark that even when $\mu _{\omega }(\varphi _{\omega })$ does not depend on ${\omega }$ other finer results like the ASIP do not follow by integration. Indeed the ASIP concerns an almost sure approximation of the partial sums in question by a sum of independent Gaussian random variables, but the quenched ASIP provides a construction of such a Gaussian process which depends on the fiber ${\omega }$ .
1.2.1 Annealed limit theorems: i.i.d. maps
A particular wellstudied case is when the maps $T_{{\sigma }^j{\omega }}$ are independent. That is, ${\Omega }={\mathcal Y}^{\mathbb Z}$ is a product space, the coordinates ${\omega }_j$ of ${\omega }=({\omega }_j)$ are independent (with ${\sigma }$ being the left shift) and $T_{\omega }=T_{{\omega }_0}$ depends only on the zeroth coordinate. In this case the statistical behavior of the skew product $\tau $ can be investigated using the socalled annealed transfer operator, given by (see [Reference Baladi8, Reference Baladi and Young9, Reference Ishitani35])
where ${\mathcal L}_{\omega }$ is the transfer operator corresponding to $T_{\omega }$ and the underlying reference measure m. In [Reference Aimino, Nicol and Vaienti2] it was shown that for several classes of random expanding maps, the operator ${\mathcal A}$ is quasicompact. Using that, a variety of limit theorems were obtained (such as a CLT, a Berry–Esseen theorem, a local CLT, a local largedeviations principle and an ASIP) for random variables of the form
where $({\omega },x)$ are distributed according to a $\tau $ invariant measure $\mu $ of the form ${{\mathbb P}\times (h\,dm)}$ for some continuous function h, which satisfies ${\mathcal A} h=h$ . The latter assumption means that the maps $T_{\omega }$ preserve the same measure $\nu =h\, dm$ . The point is that once quasicompactness is achieved the classical Nagaev–Guivarch method (see [Reference Hennion and Hervé33]) can be applied. This method was applied successfully to obtain limit theorems for deterministic dynamical systems (that is, when $T_{\omega }=T$ does not depend on ${\omega }$ ), and in [Reference Aimino, Nicol and Vaienti2] (see also [Reference Ayyer, Liverani and Stenlund7]) this method was applied to obtain annealed limit theorems. We note that since both the function $\varphi $ and the measure $h \,dm$ do not depend on ${\omega }$ , and all the maps $T_{\omega }$ preserve the measure $h \,dm$ , the fiberwise centering constant $\mu _{\omega }(S_n^{\omega }\varphi )$ and the usual centering constant $\mu (S_n\varphi )$ are both equal to $n\int \varphi (x)h(x)\,dm(x)$ . Hence, as discussed in the previous section, in this setup some annealed results such as the CLT already follow from the quenched ones.
Independence here is crucial, since it yields that the iterates on the annealed transfer operator can be written as
where ${\mathcal L}_{\omega }^n={\mathcal L}_{{\sigma }^{n1}{\omega }}\circ \cdots \circ {\mathcal L}_{{\sigma }{\omega }}\circ {\mathcal L}_{\omega }$ , which is the transfer operator of $T_{\omega }^n$ . Hence, the statistical behavior of the iterates $\tau ^n$ of the skew product can be described by the iterates of ${\mathcal A}$ . Note that in this independent and identically distributed (i.i.d.) setup this approach works only when $\varphi ({\omega },x)=\varphi (x)$ does not depend on ${\omega }$ since it requires substituting $\varphi $ (and appropriate functions of $\varphi $ ) into the annealed operator.
1.2.2 The motivation behind the present paper: noni.i.d. maps and random functions
The starting point of this paper is the observation that when the coordinates $({\omega }_j)$ are not independent (that is, that maps $T_{{\sigma }^j{\omega }}$ are not i.i.d.) there is no apparent relation between the iterates $\tau ^n$ of $\tau $ and the iterates of the annealed operator ${\mathcal A}$ defined above. Thus, a natural question arising from [Reference Aimino, Nicol and Vaienti2, Reference Ayyer, Liverani and Stenlund7] is which limit theorems hold true for mixing base maps with nonindependent coordinates, and functions $\varphi $ which depend on ${\omega }$ . Moreover, the assumptions in [Reference Aimino, Nicol and Vaienti2] require all the maps $T_{\omega }$ to preserve the same absolutely continuous measure $\nu =h\, dm$ , and it is also desirable to prove limit theorems without such assumptions. (We refer to [Reference Nicol, Pereira and Török46] for a CLT and large deviations for random i.i.d. intermittent maps in the case where the $T_{\omega }$ do not preserve the same measure.) We note that without the above assumptions even the CLT was not obtained before for the skew products considered in this paper, which will be our first result.
The question described above was also one of the main motivations in [Reference Hafouta26], where a CLT, a local CLT and a renewal theorem were obtained for several classes of skew products with mixing base maps such as Markov shifts and nonuniform Young towers, together with uniformly expanding random maps. These results were obtained by a certain type of integration argument; however, the method of [Reference Hafouta26] does not involve the iterates of an annealed transfer operator, and instead we studied directly integrals of the form $\int {\mathcal L}_{{\omega }}^ng_{\omega }\, d{\mathbb P}({\omega })$ , and their complex perturbations (relying on the fiberwise ‘spectral’ properties and a certain type of periodic point approach which was introduced in [Reference Hafouta and Kifer31]). While [Reference Hafouta26] was the first paper to discuss limit theorem for skew products with nonindependent fiber maps and random observables, all the results there were obtained for fiberwise centered observables $\varphi $ (that is, $\mu _{\omega }(\varphi _{\omega })=0$ ). Moreover, the maps $T_{\omega }$ in [Reference Hafouta26] were uniformly expanding, the base map had a periodic point and the random transfer operator satisfied certain regularity assumptions as functions of ${\omega }$ around the periodic orbit. From this point of view, a second motivation for the present paper is to prove limit theorems for skew products with nonindependent fiber maps $T_{{\sigma }^j{\omega }}$ without the fiberwise centralization assumption and without additional topological assumptions such as the behavior around a periodic orbit. We note that, apart from the CLT, we did not consider in [Reference Hafouta26] any of the limit theorems obtained in the present paper, and so almost all the results in the present paper are new even under the fiberwise centering assumption.
1.3 Our new results and the method of the proofs
As explained in the previous section, the goal of this paper is to obtain limit theorems with deterministic centering conditions for skew products $\tau $ built over mixing base maps and nonuniformly expanding maps $T_{\omega }$ . More precisely, we still consider a product space ${\Omega }={\mathcal Y}^{\mathbb Z}$ , but with ‘weakly dependent’ coordinates ${\omega }_j$ instead of independent ones. We consider a family of nonuniformly expanding maps $T_{\omega }=T_{{\omega }_0}$ and observables of the form $\varphi ({\omega },x)=\varphi _{{\omega }_0}(x)$ and prove limit theorems for sequences of the form $Z_n=S_n\varphi n\int \varphi\, d\mu $ , where
considered as a random variables on the probability space $({\Omega }\times X,{\mathcal F}\times {\mathcal B},\mu )$ , where $\mu =\int \mu _{\omega }\, d{\mathbb P}$ is the unique $\tau $ invariant measure with $\mu _{\omega }$ being absolutely continuous with respect to m (or when $\mu _{\omega }$ is a random Gibbs measure). In this setup we have $(T_{{\omega }})_*\mu _{\omega }=\mu _{{\sigma }{\omega }}$ , and in general the maps $T_{\omega }$ do not preserve the same measure. These results are obtained for a certain type of observables $\varphi $ so that $\varphi _{{\omega }}(\cdot )$ has bounded variation, uniformly in ${\omega }$ . When the maps $T_{\omega }$ are expanding on average we will also have a certain scaling assumption (that is, $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ for some tempered random variable K), which was shown in [Reference Dragičević, Hafouta and Sedro22] to be necessary for quenched limit theorems, and which is similarly necessary for obtaining limit theorems for the skew product. In what follows we will always assume that $\int \varphi\, d\mu =0$ , which is not really a restriction since we can always replace $\varphi $ with $\varphi \int \varphi\, d\mu $ .
We obtain our results using two different methods, as described below.
1.3.1 Limit theorems for skew products: (functional) CLT, moment estimates, moderatedeviations and exponential concentration inequalities for ${\alpha }$ mixing driving systems via the method of cumulants
Recall that the ${\alpha }$ mixing (dependence) coefficient between two sub ${\sigma }$ algebras ${\mathcal G},{\mathcal H}$ of ${\mathcal F}$ is given by
Let ${\mathcal F}_{\infty ,k}$ be the ${\sigma }$ algebra generated by the coordinates ${\omega }_j$ at places $j\leq k$ and ${\mathcal F}_{m,\infty }$ be the ${\sigma }$ algebra generated by the coordinates ${\omega }_j$ at places $j\geq m$ . Then the ${\alpha }$ dependence coefficients of the sequence of coordinates $({\omega }_n)$ are defined by
where the last equality is due to stationarity of the process $({\omega }_n)$ .
We assume first that ${\alpha }_n=O(e^{cn^\eta })$ for some $c,\eta>0$ (that is, it is stretched exponential). The first step towards limit theorems is standard for stationary processes: we show that under the weaker condition $\sum _{n}n{\alpha }_n<\infty $ , the limit
exists and that it vanishes if and only if $\varphi $ admits an appropriate coboundary representation. When $s^2>0$ we show that $n^{1/2}S_n$ converges in distribution towards a centered normal random variable with variance $s^2$ . More precisely, we obtain the convergence rate
An annealed CLT (that is, for independent maps) was obtained in [Reference Ayyer, Liverani and Stenlund7] for random toral automorphisms and in [Reference Aimino, Nicol and Vaienti2] for more general maps. When the base map is only mixing (and $\varphi $ depends on ${\omega }$ ) it was obtained in [Reference Hafouta26] for fiberwise centered potentials (that is, ${\mu _{\omega }(\varphi _{\omega })=0}$ ). One of the results in this paper is the CLT for stretched exponentially ${\alpha }$ mixing base maps but without the fiberwise centering assumption (in fact, we will obtain a functional CLT; see Theorem 2.19 and the last paragraph of §1.3.1).
We also obtain a certain type of largedeviations results, often referred to as a moderatedeviations principle (see [Reference Dembo and Zeitouni14]). These results yield, for instance, that for every closed interval $[a,b]$ we have
where $a_n$ is a sequence such that $a_n\to \infty $ and $a_n=o(n^{{1}/({2+4{\gamma }})})$ . We also obtain several types of ‘stretched’ exponential concentration inequalities ((2.20), (2.21)) and Gaussian moment estimates of Rosenthal type (2.22). These result are obtained using the method of cumulants. More precisely, we first obtain a certain type of multiple correlation estimates (see Proposition 3.4), and then by applying a general theorem we conclude that the kth cumulant of the sum $S_n$ is at most of order $ n(k{!})^{1+{\gamma }}(c_0)^{k2}$ for $k\geq 3$ , where $c_0$ is some constant (see Theorem 3.1). Then we can apply the method of cumulants [Reference Döring and Eichelsbacher15, Reference Saulis and Statulevicius49]. In the annealed setup, using the quasicompactness of the annealed transfer operator, largedeviations principles and exponential concentration inequalities were obtained in [Reference Aimino, Nicol and Vaienti2], and the above results show that there is a similar behavior when the maps are not independent and the function $\varphi $ depends on ${\omega }$ (see also the results in the next section where better exponential concentration inequalities are described).
The above multiple correlation estimates together with the method of cumulants and the Rosenthaltype moment estimates also yield a functional CLT. Let us consider the random function ${\mathcal S}_n(t)=n^{1/2}S_{[nt]}$ on $[0,1]$ . Then we show that it converges in distribution in the Skorokhod space $D[0,1]$ to $sW$ , where W is a standard Brownian motion and $s^2=\lim _{n\to \infty }( 1/n)\mathrm {Var}_\mu (S_n)$ .
1.3.2 Limit theorems for skew products with $\phi $  or $\psi $ mixing driving systems via martingale methods: almost sure invariance principle, concentration inequalities and maximal moment estimates
One of the strongest methods to prove CLTs and related results in probability theory and dynamical systems is the socalled martingalecoboundary representation (Gordin’s method). For a sufficiently chaotic dynamical system $(Y,{\mathcal G},\mu ,T)$ and an observable $\varphi :Y\to {\mathbb R}$ it means that $\varphi $ can be represented as $\varphi =u+\chi \chi \circ T$ for some sufficiently regular function $\chi $ , and $(u\circ T^n)$ forms a reverse martingale difference. Such results are well known for deterministic expanding (or hyperbolic) dynamical systems, and we refer to [Reference Dragičević, Froyland, GonzálezTokman and Vaienti16, Reference Dragičević, Hafouta and Sedro22, Reference Korepanov, Kosloff and Melbourne42] for quenched and sequential versions of such martingale methods.
Recall that the $\phi $ mixing and $\psi $ (dependence) coefficient between two sub ${\sigma }$ algebras ${\mathcal G},{\mathcal H}$ of ${\mathcal F}$ is given by
and
The reverse $\phi $ mixing coefficients of the sequence of coordinates $({\omega }_n)$ are defined by
while the $\psi $ mixing coefficients of $(\xi _n)$ are defined by
where ${\mathcal F}_{n,m}$ is as defined before (1.2). It is clear from the definitions of the mixing coefficients that
When the sequence $({\omega }_n)$ is (sufficiently fast) $\phi $  or $\psi $ mixing we obtain a certain type of $L^\infty $ martingalecoboundary representation (that is, $\chi \in L^\infty $ ) for the underlying class of observables $\varphi $ with respect to the skew product $\tau $ . This was already established in [Reference Aimino, Nicol and Vaienti2] in the annealed setup (that is, when $({\omega }_n)$ is an i.i.d. sequence), and here, using different arguments, we obtain such a representation for skew products with mixing base maps.
Once an $L^\infty $ martingalecoboundary decomposition is achieved, as usual, we can apply the Azuma–Hoeffding inequality together with Chernoff’s bounding method and obtain exponential concentration inequalities of the form
where $c_1,c_2,c_3$ are positive constants. These concentration inequities are better than the ones we obtain using the method of cumulants, although they involve the stronger notions of $\phi $  or $\psi $ mixing instead of ${\alpha }$ mixing. (However, they only require summable $\phi $  or $\psi $ mixing coefficients and not stretched exponential ones.) Another immediate consequence is moment estimates of the form
which hold for every $1\leq p<\infty $ . Such results are known in the annealed case [Reference Aimino, Nicol and Vaienti2], and we extend them to the skew products considered in this paper.
The idea behind the martingalecoboundary representation is as follows. Consider the sub ${\sigma }$ algebra ${\mathcal F}_0$ of ${\Omega }\times X$ generated by the projection $\pi _0({\omega },x)=(({\omega }_j)_{j\geq 0},x)$ , where ${\omega }=({\omega }_j)_{j\in {\mathbb Z}}$ . Then $\tau $ preserves ${\mathcal F}_0$ since $T_{\omega }=T_{{\omega }_0}$ depends only on ${\omega }_0$ , and ${\mathcal F}_0$ can be viewed as a subsystem (or a factor) given by $({\Omega }\times X,{\mathcal F}_0,\mu ,\tau )$ . Our main argument is as follows. Let ${\mathcal K}$ be the transfer operator corresponding to the invariant ${\sigma }$ algebra ${\mathcal F}_0$ , namely the one defined by the duality relation
where $g\in L^1({\Omega }\times X,{\mathcal F}_0,\mu )$ and $f\in L^\infty ({\Omega }\times X,{\mathcal F}_0,\mu )$ . Then we show that, under quite mild $\phi $  or $\psi $ mixing rates for the sequence of coordinates $({\omega }_n)$ , the iterates ${\mathcal K}^{\,n} \varphi $ of the transfer operator ${\mathcal K}$ corresponding to this system converge fast enough in $L^\infty (\mu )$ towards $\mu (\varphi )\textbf {1}$ , where $\textbf {1}$ is the function taking the constant value $1$ , and $\varphi $ is our given observable. This convergence can be established for every function $\varphi $ so that $\\varphi \_{K,2}=\text {esssup}_{{\omega }\in {\Omega }}(K({\omega })^2\\varphi ({\omega },\cdot )\_{BV})<\infty $ for an appropriate tempered random variable $K({\omega })$ , or for any observable with $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}\\varphi ({\omega },\cdot )\_{BV}<\infty $ when the maps $T_{\omega }$ are uniformly expanding. We stress that in any case this is not a spectral result (even under exponential mixing), since the convergence of ${\mathcal K}^{\,n}$ is not in an operator norm, and, in general, it does not have an exponential rate. Indeed, we only prove that
where $\gamma _n={\delta }^n+\phi _R([n/2])$ or $\gamma _n={\delta }^n+\psi ([n/2])$ , and ${\delta }\in (0,1)$ and $\phi _R(\cdot )$ and $\psi (\cdot )$ are the reverse $\phi $ mixing coefficients and $\psi $ mixing coefficients defined in (1.3) and (1.4), respectively.
Another consequence of the martingalecoboundary 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, [Reference Dragičević, Froyland, GonzálezTokman and Vaienti16, Reference Dragičević, Hafouta, Pollicott and Vaienti20–Reference Dragičević, Hafouta and Sedro22, Reference Haydn, Nicol, Török and Vaienti32, Reference Su50, Reference Su51]), and in this paper we will focus on the ASIP for Birkhoff sums generated by the skew product.
In [Reference Cuny and Merlevede13] the authors proved that, under certain assumptions, a reverse martingale $M_n$ can be approximated almost surely by a sum of independent Gaussians. One consequence of the methods in [Reference Cuny and Merlevede13] is for sums of the form $W_n=\sum _{j=0}^{n1}\varphi \circ \tau ^j$ . For such sums, the conditions of [Reference Cuny and Merlevede13, Theorem 3.2] show that there is a coupling with a sequence of i.i.d. centered normal random variables $Z_j$ with variance $s^2=\lim _{n\to \infty }(1/n)\mathrm {Var}(W_n)$ so that
In our notation, the first and second conditions of [Reference Cuny and Merlevede13, Theorem 3.2] about ${\mathcal K}$ can be verified using (1.5). In order to show that the third (and last condition) about ${\mathcal K}$ in [Reference Cuny and Merlevede13, Theorem 3.2] is in force we will also need to provide more general estimates on expression of the form
for $1\leq i,j\leq n$ , where $\bar \varphi =\varphi \mu (\varphi )$ .
We note that in [Reference Aimino, Nicol and Vaienti2] the annealed ASIP was obtained using Gouëzel’s approach [Reference Gouëzel24] and not the martingalecoboundary approach. Gouëzel’s approach was also used in [Reference Atnip5] to obtain an ASIP for nonindependent maps with mixing base maps, but as indicated in [Reference Atnip5] the results are mostly applicable for Gordin–Denker maps.
Finally, we also prove a vectorvalued ASIP for skew products with uniformly expanding random maps and exponentially fast ${\alpha }$ mixing base maps via the method of Gouëzel [Reference Gouëzel24]. As we have mentioned above, this method was applied in [Reference Aimino, Nicol and Vaienti2] in the annealed setting, while in [Reference Atnip5] it was applied for Gordin–Denker systems. In a final section we also discuss a few extensions such as different types of mixing base maps such as Young towers or Gibbs–Markov maps, application of the method of cumulants for nonconventional sums of the form $S_n=\sum _{m=1}^n\prod _{j=1}^\ell \varphi _j\circ \tau ^{q_j(m)}$ , for polynomial $q_j(m)$ , as well as extensions of the results for different classes of random expanding maps (the ones in [Reference Mayer, Skorulski and Urbański44]).
2 Preliminaries and main results
2.1 The random maps
We begin by recalling the setup from [Reference Buzzi12]. Let $(X, \mathcal G)$ be a measurable space endowed with a probability measure m and a notion of a variation $\mathrm {v} \colon L^1(X, m) \to [0, \infty ]$ which satisfies the following conditions:

(V1) $\mathrm {v} (th)= t \mathrm {v} (h)$ ;

(V2) $\mathrm {v} (g+h)\leq \mathrm {v} (g)+\mathrm {v} (h)$ ;

(V3) $\h\_{L^\infty } \le C_{\mathrm {v}}(\h\_1+\mathrm {v} (h))$ for some constant $1\leq C_{\mathrm {v}}<\infty $ ;

(V4) for any $C>0$ , the set $\{h\colon X \to \mathbb R: \h\_1+\mathrm {v} (h) \leq C\}$ is $L^1(m)$ compact;

(V5) $\mathrm {v}({\boldsymbol 1})=0$ , where ${\boldsymbol 1}$ denotes the function equal to $1$ on X;

(V6) $\{h \colon X \to \mathbb R_+: \lVert h\rVert _1=1 \ \text {and} \ \mathrm {v} (h)<\infty \}$ is $L^1(m)$ dense in $\{h\colon X \to \mathbb R_+: \h\_1=1\}$ ;

(V7) for any $f\in L^1(X, m)$ such that $\text {essinf} f>0$ , we have
$$ \begin{align*}\mathrm{v}(1/f) \leq \frac{\mathrm{v} (f)}{(\text{essinf} f)^2};\end{align*} $$ 
(V8) $\mathrm {v} (fg)\leq \f\_{L^\infty }\cdot \mathrm {v}(g)+\g\_{L^\infty }\cdot \mathrm {v}(f)$ ;

(V9) for $M\kern1pt{>}\kern1pt0$ , $f\kern1pt{\colon}\kern1pt X \kern1pt{\to}\kern1pt [M, M]$ measurable and every $C^1$ function $h\kern0.1pt{\colon}\kern0.1pt [M, M] \kern0.5pt{\to}\kern0.5pt {\mathbb C}$ , we have $\mathrm {v}(h\circ f)\leq \h'\_{L^\infty } \cdot \mathrm {v}(f)$ .
We define
Then $BV$ is a Banach space with respect to the norm
Remark 2.1. Observe that (V3) and (V8) imply that
Remark 2.2. We observe that in [Reference Buzzi12], assumption (V5) is replaced by the weaker . However, for the examples we have in mind, our stronger version is satisfied. In particular, (V5) implies that $\ {\boldsymbol 1}\_{BV}=1$ .
The rest of our setup is almost identical to [Reference Dragičević, Hafouta and Sedro22], with a single additional requirement that will be indicated in what follows. Let $(\Omega , \mathcal {F}, \mathbb P, \sigma )$ , be a probability space and ${\sigma \colon \Omega \to \Omega }$ an invertible ergodic measurepreserving transformation. Let $T_{\omega } \colon X \to X$ , $\omega \in \Omega $ be a collection of nonsingular transformations (that is, $m\circ T_\omega ^{1}\ll m$ for each $\omega $ ) acting on X. Each transformation $T_{\omega }$ induces the corresponding transfer operator $\mathcal L_{\omega }$ acting on $L^1(X, m)$ and defined by the duality relation
Thus, we obtain a cocycle of transfer operators $(\Omega , \mathcal F, \mathbb P, \sigma , L^1(X, m), \mathcal L)$ that we denote by $\mathcal L=(\mathcal L_\omega )_{\omega \in \Omega }$ . For $\omega \in \Omega $ and $n\in \mathbb N$ , set
We recall the notion of a tempered random variable.
Definition 2.3. We say that a measurable map $K\colon \Omega \to (0, +\infty )$ is tempered if
In this paper we will consider the following assumptions on the random transfer operators.
Definition 2.4. A cocycle $\mathcal L=(\mathcal L_\omega )_{\omega \in \Omega }$ of transfer operators is said to be good if the following conditions hold.

• $\Omega $ is a Borel subset of a separable, complete metric space and $\sigma $ is a homeomorphism. Moreover, $\mathcal L$ is $\mathbb P$ continuous, that is, $\Omega $ can be written as a countable union of measurable sets such that $\omega \mapsto \mathcal L_\omega $ is continuous on each of those sets.

• There is a tempered random variable $N({\omega })$ such that
(2.3) $$ \begin{align} v(g\circ T_\omega) \le N(\omega) v(g) \quad \text{for }{\mathbb P}\text{a.e. }\omega \in \Omega\text{ and }g\in BV. \end{align} $$ 
• There exists a random variable $C\colon \Omega \to (0, +\infty )$ such that $\log C\in L^1(\Omega , \mathbb P)$ and
$$ \begin{align*} \\mathcal L_\omega h\_{BV}\le C(\omega) \h\_{BV} \quad \text{for }{\mathbb P}\text{a.e. }\omega \in \Omega\text{ and }h\in BV. \end{align*} $$ 
• There exist $N\in {\mathbb N}$ and random variables $\alpha , K \colon \Omega \to (0, +\infty )$ such that
$$ \begin{align*} \int_\Omega \log \alpha \, d\mathbb P <0, \quad \log K \in L^1(\Omega, \mathbb P) \end{align*} $$and, for $\mathbb P$ a.e. $\omega \in \Omega $ and $h\in BV$ ,$$ \begin{align*} \mathrm{v}(\mathcal L_\omega^N h) \leq \alpha (\omega) \mathrm{v}(h)+ K(\omega) \h\_1. \end{align*} $$ 
• For each $a>0$ and $\mathbb P$ a.e. $\omega \in \Omega $ , there exist random numbers $n_c(\omega )<+\infty $ and $\alpha _0(\omega ), \alpha _1(\omega ), \ldots $ such that for every $h\in \mathcal C_a$ ,
(2.4) $$ \begin{align} \text{essinf}_x (\mathcal L_\omega^n h)(x) \ge \alpha_n\h \_1 \quad \text{for }n\ge n_c, \end{align} $$where(2.5) $$ \begin{align} \mathcal C_a:=\{ h\in L^\infty (X,m): h\geq 0 \text{ and }\mathrm{v}(h) \leq a\h\_1\}. \end{align} $$ 
• .
Finally, we say that the cocycle $\mathcal L$ is uniformly random if the random variables $C,{\alpha }^N, K^N$ and $n_c$ are constants and ${\alpha }_n({\omega })$ does not depend on n and ${\omega }$ .
Remark 2.5

• Definition 2.4 almost coincides with [Reference Dragičević, Hafouta and Sedro22, Definition 3], the only difference being the addition of (2.3) (which was considered in [Reference Dragičević, Hafouta and Sedro22, §3].)

• The logintegrability assumption specified at the end of Definition 2.4 may easily be checked on explicit examples (see, for example, the discussion in [Reference Atnip, Froyland, GonzálezTokman and Vaienti6, Remark 2.12]).

• Furthermore, this assumption implies a certain version of the ‘random covering’ similar to (2.4); see [Reference Dragičević, Hafouta and Sedro22, Remark 4].
Let us now give examples of systems satisfying our requirements. Our first example is essentially taken from [Reference Buzzi12].
Example 2.6. (Lasota–Yorke cocycles)
Consider $X=[0,1]$ , endowed with Lebesgue measure m and the classical notion of variation $\mathrm {v}$ . We say that $T:X\to X$ is a piecewise monotonic nonsingular (p.m.n.s.) map if the following conditions hold.

• T is piecewise monotonic, that is, there exists a subdivision $0=a_0<a_1<\cdots <a_N=1$ such that for each $i\in \{0,\ldots ,N1\}$ , the restriction $T_i=T_{(a_i,a_{i+1})}$ is monotonic (in particular, it is a homeomorphism on its image).

• T is nonsingular, that is, there exists $T':[0,1]\to \mathbb R_+$ such that, for any measurable $E\subset (a_i,a_{i+1})$ , $m(T(E))=\int _ET'\,dm$ .
The intervals $(a_i,a_{i+1})_{i\in \{0,\ldots ,N1\}}$ are called the intervals of T. We also set $N(T):=N$ and $\unicode{x3bb} (T):=\text {essinf }_{[0,1]}T'$ .
We consider a family $(T_\omega )_{\omega \in \Omega }$ of random p.m.n.s. as above, and such that $T:\Omega \times [0,1]\to [0,1], (\omega ,x)\mapsto T_\omega (x)$ is measurable. Denoting $N_\omega =N(T_\omega )$ and $\unicode{x3bb} _\omega =\unicode{x3bb} (T_\omega )$ , we make the following assumptions.

• The map $\omega \mapsto (\mathrm {v}({1}/{T^{\prime }_\omega }),N_\omega ,\unicode{x3bb} _\omega ,a_1,\ldots ,a_{N_\omega 1})$ is measurable.

• We have the following expandingonaverage property:

• The maps $\log (N_{\omega })$ and $\log ^+({N_\omega }/{\unicode{x3bb} _\omega })$ are integrable.

• The map $\log ^+(\mathrm {v}({1}/{T_\omega '}))$ is integrable.

• $T_\omega $ is covering, that is, for any interval $I\subset [0,1]$ , there exists a random number $n_c(\omega )>0$ such that, for any $n\ge n_c$ , one has
(2.6) 
• .
We will call a cocycle satisfying the previous assumptions an expandingonaverage Lasota–Yorke cocycle. For a countably valued measurable family $(T_\omega )_{\omega \in \Omega }$ of expandingonaverage Lasota–Yorke cocycles, the associated cocycle of transfer operators $(\mathcal L_\omega )_{\omega \in \Omega }$ is good (see [Reference Dragičević and Sedro23]).
The following example can be fruitfully compared to a similar one by Kifer [Reference Kifer38].
Example 2.7. We consider $X=\mathbb S^1$ , endowed with the Lebesgue measure m and the notion of variation given by $\mathrm {v}(\phi ):=\int _X \phi '~dm=\\phi '\_{L^1}$ . We consider a measurable map $T:\Omega \times X\to X$ such that $T_\omega :=T(\omega ,\cdot )$ is $C^r$ , $r\ge 2$ . In addition, we make the following assumptions.

• There exists a tempered random variable $N({\omega })$ so that (2.3) holds true.

• The map $\omega \in \Omega \mapsto (\int _X({T_\omega "}/{(T_\omega ')^2}) \,dm,\unicode{x3bb} _\omega )$ is measurable, where $\unicode{x3bb} _\omega =\inf _{[0, 1]}T_\omega '$ .

• The following expandingonaverage property holds:
(2.7) 
• The map $\log (\int _X{T_\omega "}/{(T_\omega ')^2}\, dm)$ is $\mathbb P$ integrable.

• .
We call a family $(T_\omega )_{\omega \in \Omega }$ satisfying the previous assumptions a smooth expandingonaverage cocycle. For a family $(T_\omega )_{\omega \in \Omega }$ , countably valued and measurable, of smooth expandingonaverage cocycles which satisfy (2.3), the associated cocycle of transfer operators $(\mathcal L_\omega )_{\omega \in \Omega }$ is good (see [Reference Dragičević and Sedro23, Example 16]). We note that our expansiononaverage condition (2.7) implies that $\mathbb P$ almost surely, $T_\omega $ has nonvanishing derivative, hence is a local diffeomorphism and a monotonic map of the circle. As noted in [Reference Dragičević, Hafouta and Sedro22, Example 6], smooth expandingonaverage cocycles satisfy a stronger version of the random covering property (which by [Reference Buzzi12, Remark 0.1] implies the one formulated in (2.6)): for each nontrivial interval $I\subset X$ , for $\mathbb P$ a.e. $\omega \in \Omega $ , there is an $n_c:=n_c(\omega ,I)<\infty $ such that, for all $n\ge n_c$ ,
2.2 The onedimensionality of the top Oseledets space: a summary of known results
In this section we recall two results from [Reference Dragičević, Hafouta and Sedro22] that will be in constant use in the course of the proofs of all of our results.
Theorem 2.8. [Reference Dragičević, Hafouta and Sedro22, Theorem 12]
Let $\mathcal L=(\mathcal L_\omega )_{\omega \in \Omega }$ be a good cocycle of transfer operators. Then the following assertions hold.

• There exists an essentially unique measurable family $(h_\omega )_{\omega \in \Omega }\subset BV$ such that $h_\omega \ge 0$ , $\int _X h_\omega \, dm=1$ and
$$ \begin{align*} \mathcal L_\omega h_\omega=h_{\sigma \omega} \quad \text{for }{\mathbb P}\text{a.e. }\omega \in \Omega. \end{align*} $$ 
• There is a random variable $\ell :\Omega \to (0,+\infty )$ such that, for $\mathbb P$ a.e. $\omega \in \Omega $ ,
(2.8) $$ \begin{align} h_\omega \ge \ell(\omega), \quad m\text{a.e.} \end{align} $$ 
• For $\mathbb P$ a.e. $\omega \in \Omega $ ,
(2.9) $$ \begin{align} BV=\mathrm{span}\{h_\omega\} \oplus BV^0, \end{align} $$where$$ \begin{align*} BV^0=\bigg \{ h\in BV: \int_X h\, dm=0 \bigg \}. \end{align*} $$ 
• $\omega \mapsto \h_\omega \_{BV}$ is tempered.

• There exist $\unicode{x3bb}>0$ and for each $\epsilon>0$ , a tempered random variable $D=D_\epsilon \colon \Omega \to (0, +\infty )$ such that, for $\mathbb P$ a.e. $\omega \in \Omega $ and $n\in \mathbb N$ ,
(2.10) $$ \begin{align} \ \mathcal L_\omega^n \Pi(\omega) \_{BV} \le D(\omega)e^{\unicode{x3bb} n} \end{align} $$and(2.11) $$ \begin{align} \ \mathcal L_\omega^n(\mathrm{Id} \Pi(\omega)) \_{BV} \le D(\omega)e^{\epsilon n}, \end{align} $$where $\Pi (\omega ) \colon BV \to BV^0$ is a projection associated to the splitting (2.9).Finally, for uniformly random cocycles the random variables $\ell ({\omega })$ and $D({\omega })$ can be replaced with positive constants and ${\omega }\to \h_{\omega }\_{BV}$ is a bounded random variable.
Corollary 2.9. [Reference Dragičević, Hafouta and Sedro22, Corollary 13]
Let $\mathcal L=(\mathcal L_\omega )_{\omega \in \Omega }$ be a good cocycle of transfer operators. Then the following assertions hold.

• If $(h_\omega )_{\omega \in \Omega }\subset BV$ is given by Theorem 2.8, then
(2.12) $$ \begin{align} \omega \mapsto \ 1/h_\omega\_{BV} \ \text{is tempered.} \end{align} $$ 
• For $\mathbb P$ a.e. $\omega \in \Omega $ ,
(2.13)where$$ \begin{align*} BV_\omega^0=\bigg \{ h\in BV: \int_X h\, d\mu_\omega=0 \bigg \}, \end{align*} $$in which $d\mu _\omega =h_\omega dm$ , $\omega \in \Omega $ ; 
• there exist $\unicode{x3bb} '>0$ and a tempered random variable $\tilde D\colon \Omega \to (0, +\infty )$ such that, for $\mathbb P$ a.e. $\omega \in \Omega $ and $n\in \mathbb N$ ,
(2.14) $$ \begin{align} \ L_\omega^n \tilde \Pi(\omega) \_{BV} \le \tilde D(\omega)e^{\unicode{x3bb}' n}, \end{align} $$(2.15) $$ \begin{align} \ L_\omega^n(\mathrm{Id} \tilde \Pi(\omega)) \_{BV} \le \tilde D(\omega), \end{align} $$where $\tilde \Pi (\omega ) \colon BV \to BV_\omega ^0$ is a projection associated to the splitting (2.13), and$$ \begin{align*} L_\omega^n g =\mathcal L_\omega^n(g h_\omega) /h_{\sigma^n \omega}, \quad g\in BV, \ n\in \mathbb N. \end{align*} $$Finally, for uniformly random cocycles the random variable $\tilde D({\omega })$ can be replaced with a positive constant.
Since ${\mathcal L}_{\omega } h_{\omega }=h_{\sigma {\omega }}$ and ${\mathcal L}_{\omega }$ satisfy the duality relation (2.2), the measure $\mu _{\omega }$ satisfies that for ${\mathbb P}$ a.e. ${\omega }$ we have $(T_{\omega })_*\mu _{\omega }=\mu _{{\sigma }_{\omega }}$ . Thus, if $\tau :{\Omega }\times X\to {\Omega }\times X$ is defined by $\tau ({\omega },x)=({\sigma }{\omega },T_{\omega } x)$ then $\mu _{\omega }$ gives rise to a $\tau $ invariant probability measure $\mu $ on ${\Omega }\times X$ such that
for every measurable set A in ${\Omega }$ and B in X, where $h({\omega },x)=h_{\omega }(x)$ .
2.3 Main results: limit theorems for mixing base maps
2.4 The observable
Let us take a measurable $\varphi :{\Omega }\times X\to {\mathbb R}$ so that $\int \varphi\, d\mu =0$ . Let $\tilde K({\omega })$ be the tempered random variable defined by
where $D({\omega }),\tilde D({\omega })$ and $N({\omega })$ are specified in the definition of a good cocycles and in Theorem 2.8 and Corollary 2.9. In order to describe our assumptions on the observable $\varphi $ , we will need the following classical result (see [Reference Arnold4, Proposition 4.3.3.]).
Proposition 2.10. Let $\tilde K \colon \Omega \to (0, +\infty )$ be a tempered random variable. For each $\epsilon>0$ , there exists a tempered random variable $\tilde K_\epsilon \colon \Omega \to (1, +\infty )$ such that
for $\mathbb P$ a.e. $\omega \in \Omega $ and $n \in \mathbb Z$ .
Next, using the notation of Proposition 2.10, let $K({\omega })=\tilde K_{\varepsilon }({\omega })$ for some ${\varepsilon }<{\unicode{x3bb} }"/3$ , where ${\unicode{x3bb} }"=\min ({\unicode{x3bb} },{\unicode{x3bb} }')$ , and ${\unicode{x3bb} }$ and ${\unicode{x3bb} }'$ are specified in Theorem 2.8 and Corollary 2.9, respectively.
Remark 2.11. From now on we will replace both ${\unicode{x3bb} }$ and ${\unicode{x3bb} }'$ by their minimum, which for notational convenience will be denoted by ${\unicode{x3bb} }$ .
In what follows we will consider an observable $\varphi :{\Omega }\times X\to {\mathbb R}$ satisfying the scaling condition
which was first introduced in [Reference Dragičević and Sedro23]. In the uniformly random case $\tilde K({\omega })$ (and hence $K({\omega })$ ) can be replace by a positive constant, and so the scaling condition reads
The main goal in this paper is to obtain limit theorems for the sequence of functions
under certain mixing assumptions on the driving system $({\Omega },{\mathcal F},{\mathbb P},\sigma )$ and the above assumptions on the observable $\varphi $ .
Remark 2.12. For expandingonaverage maps the scaling condition (2.16) is necessary for limit theorems (see [Reference Dragičević, Hafouta and Sedro22, Appendix]). In any case, our results are also new in the uniformly random case, and readers who would prefer can just consider this case together with the assumption that $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}\\varphi _{\omega }\_{BV}<\infty $ .
Let us also note that, in general, the random variable $K({\omega })$ comes from Oseledets theorem and it is not computable. In order to provide explicit conditions for quenched limit theorems, in [Reference Hafouta28] several examples of nonuniformly expanding maps (which are stronger than expansion on average) were given with the property that
Here the $BV$ norm is with respect to the choice of variation $v(g)=v_{\alpha }(g)$ , where $v_{\alpha }$ is the Hölder constant corresponding to some exponent ${\alpha }$ and $B({\omega })$ and $\rho ({\omega })\in (0,1)$ are random variables with explicit formulas, and they depend only on the zeroth coordinate ${\omega }_0$ . Moreover, for several of these examples we already have $B({\omega })\leq B$ for some constant B. In this case (similarly to [Reference Dragičević, Hafouta and Sedro22, §5.2]) we have the following assertions. Let ${\varepsilon }$ be smaller than $1{\mathbb E}_{\mathbb P}[\rho ]$ and let $A=\{{\omega }: \rho ({\omega })\leq 1{\varepsilon }\}$ . Then ${\mathbb P}(A)>0$ . Let ${R_n({\omega })=\sum _{j=0}^{n1}{\mathbb I}({\sigma }^j{\omega }\in A)}$ . Then $R_n/n\to r={\mathbb P}(A)$ ( ${\mathbb P}$ almost surely). Let
Then, for $\mathbb P$ a.e. $\omega \in \Omega $ and $n\in {\mathbb N}$ ,
where $K(\omega )=B(1{\varepsilon })^{N(\omega )}$ . Observe that for $k\ge 1$ ,
Thus, if the stationary process $(\mathbb I_A\circ \sigma ^n)$ satisfies an appropriate concentration inequality (for example, under appropriate mixing assumptions on $(\xi _n)$ ), we can conclude that $N(\omega )$ is integrable. Hence, $\log K$ is integrable and consequently also tempered.
The above means that in this situation we can express the condition on $\varphi $ by means of the more explicit random variable $K({\omega })$ defined above. Still, in the setup of [Reference Hafouta28], under appropriate integrability conditions on $B({\omega })$ the main results in this paper can be obtained under conditions such as $\varphi \in L^p(\mu )$ for p large enough (depending on the desired result). Since this approach requires several nontrivial modifications to the arguments in this paper such results will be considered elsewhere.
2.5 Limit theorems
Let us first introduce our assumptions on the base map. Let $(\xi _n)$ be a twosided stationary sequence taking values on some measurable space ${\mathcal Y}$ . We assume here that $({\Omega },{\mathcal F},{\mathbb P},\sigma )$ is the corresponding shift system. Namely, $\Omega ={\mathcal Y}^{\mathbb Z}$ , $({\sigma } {\omega })_j=({\omega }_{j+1})_j$ is the left shift and if $\pi _0:\Omega \to {\mathcal Y}$ denotes the zeroth coordinate projection, then $(\xi _n)$ has the same distribution as $(\pi _0\circ \sigma ^n)$ . We also assume that $T_{\omega }=T_{{\omega }_0}$ and $\varphi ({\omega },\cdot )=\varphi ({\omega }_0,\cdot )$ depend only on zeroth coordinate ${\omega }_0$ of ${\omega }$ .
2.5.1 Limit theorems for stretched exponentially fast $\alpha $ mixing driving processes
Let $({\Omega }_0,\mathscr F,\textbf {P})$ be the probability space on which $(\xi _n)$ is defined. We recall that the ${\alpha }$ mixing (dependence) coefficient between two sub ${\sigma }$ algebras ${\mathcal G},{\mathcal H}$ of $\mathscr F$ is given by
The ${\alpha }$ dependence coefficients of $(\xi _n)$ are defined by
where $\mathscr F_{\infty ,k}$ is the $\sigma $ algebra generated by $\xi _j$ , $j\leq k$ , and $\mathscr F_{k+n,\infty }$ is generated by ${\xi _j, j\geq k+n}$ . The last equality holds true due to stationarity. Let us consider the following class of mixing assumptions on the base map.
Assumption 2.13. (Stretched exponential $\alpha $ mixing rates)
There exist positive constants $c_1,c_2$ and $\eta $ such that ${\alpha }_n\leq c_1e^{c_2n^\eta }$ for every n.
Our first result concerns the variance of $S_n$ and the CLT (with rates).
Theorem 2.14. Suppose that the cocycle ${\mathcal L}$ is good. Let $\varphi $ be an observable such that $\\varphi \_K:=\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ , where $\varphi _{\omega }=\varphi ({\omega },\cdot )$ . Suppose that ${\sum _n n{\alpha }_n<\infty }$ . Then the limit
exists and vanishes if and only if $\varphi =r\circ \tau r$ for some $r\in L^2(\mu )$ . If in addition Assumption 2.13 is satisfied then $n^{1/2}S_n$ converges in distribution to $sZ$ , where Z is a standard normal random variable. Moreover, there is a constant $C>0$ such that. for all $n\in {\mathbb N}$ ,
where $\gamma =1/\eta $ and $\Phi $ is the standard normal distribution function. The constant C depends only on $c_1,c_2,\eta $ , $\\varphi \_K$ and the constant $C_{\mathrm {v}}$ (from the definition of the variation $\mathrm {v}(\cdot )$ ), and an explicit formula for C can be recovered from the proof.
The proof of Theorem 2.14 appears in §3.2.1. As discussed in §§1.2 and 1.3, when the quenched CLT holds true with a deterministic centering, then the CLT for the skew product follows by integration. This was the approach for the CLT in [Reference Aimino, Nicol and Vaienti2], but in the setup of this paper the function $\varphi $ and the measure $\mu _{\omega }$ depend on ${\omega }$ , and so the quenched CLT only holds with fiberwise centering. Thus, the novelty of Theorem 2.14 is that the CLT is obtained for the skew product beyond the annealed case considered in [Reference Aimino, Nicol and Vaienti2]. Moreover, Theorem 2.19 also strengthens the CLT in [Reference Hafouta26], since our maps $T_{\omega }$ are not uniformly expanding, and the observable $\varphi $ is not fiberwise centered.
Next, let us discuss our results concerning moderatedeviations and exponential concentration inequalities.
Theorem 2.15. Suppose that the cocycle ${\mathcal L}$ is good, and let $\varphi $ be an observable so that $\\varphi \_K=\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ . Let Assumption 2.13 hold and set ${\gamma }=1/\eta $ . Then there exist constants $a_1,a_2>0$ such that, for every $x>0$ and $n\in {\mathbb N}$ ,
All the constants depend only on $c_1,c_2,\eta $ , $\\varphi \_K$ and $C_{\mathrm {v}}$ from the definition of the variation $\mathrm {v}(\cdot )$ , and an explicit formula for them can be recovered from the proof.
We will also prove the following theorem.
Theorem 2.16. Suppose that the cocycle ${\mathcal L}$ is good, and let $\varphi $ be an observable such that $\\varphi \_K=\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ . Let Assumption 2.13 hold and set ${\gamma }=1/\eta $ . Let us also assume that the asymptotic variance $s^2$ is positive.
(i) Set $v_n=\sqrt {\mathrm {Var}(S_n)}$ , and when $v_n>0$ also set $Z_n=({S_n{\mathbb E}[S_n]})/{v_n}$ . Let $\Phi $ be the standard normal distribution function. Then there exist constants $s_3,s_4,s_5>0$ such that, that for every $n\geq a_3$ we have $v_n>0$ , and for every $0\leq x<a_4 n^{1/({2+4{\gamma }})}$ ,
The constants $a_4,a_5$ depend only on $c_1,c_2,\eta $ , $\\varphi \_K$ and $C_{\mathrm {v}}$ , and an explicit formula for them can be recovered from the proof.
(ii) Let $a_n$ , $n\geq 1$ , be a sequence of real numbers so that
Then the sequence $W_n=(sn^{1/2}a_n)^{1}S_n$ , $n\geq 1$ , satisfies the moderatedeviations principle with speed $s_n=a_n^2$ and the rate function $I(x)={x^2}/2$ . Namely, for every Borel measurable set $\Gamma \subset {\mathbb R}$ ,
where $\Gamma ^o$ is the interior of $\Gamma $ and $\overline {\Gamma }$ is its closure.
We also obtain the following Rosenthaltype moment estimates.
Theorem 2.17. Suppose that ${\mathcal L}$ is a good cocycle. If $\\varphi \_K\kern1pt{=}\kern1pt\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV}) \kern1pt{<}\infty $ , then under Assumption 2.13 there exists a constant $c_0$ such that, with $\gamma =1/\eta $ for every integer $p\geq 1$ , we have
where Z is a standard normal random variable. In particular, $\S_n{\mathbb E}_\mu [S_n]\_{L^p}=O(\sqrt n)$ for every p. As in the previous theorems, the constant $c_0$ depends (explicitly) only on $c_1,c_2,\eta $ , $\\varphi \_K$ and $C_{\mathrm {v}}$ .
We remark that Theorem 2.17 provides another proof of the CLT by the method of moments. Indeed, if $s^2>0$ then it follows that, for every integer $p\geq 1$ , the pth moment of $(S_n{\mathbb E}[S_n])n^{1/2}s^{1}$ converges to ${\mathbb E}[Z^p]$ , where $s^2$ is the asymptotic variance. In fact, for even p we get the convergence rate $O(n^{1/2})$ , while for odd p we get the rate $O(n^{1})$ .
Remark 2.18. The proofs of Theorems 2.15–2.17 appear in §3.2.2.
Theorems 2.15–2.17 are well established for sufficiently fast mixing (in the probabilistic sense) sequences of random variables, where one of the most notable methods of proof is the socalled method of cumulants (see [Reference Saulis and Statulevicius49]). For random dynamical systems, a moderatedeviations principle was obtained in [Reference Dragičević and Hafouta19], using a random complex Perron–Frobenius theorem. In the setup of [Reference Aimino, Nicol and Vaienti2], annealed (local) largedeviations principles and exponential concentration inequalities were obtained for i.i.d. maps, and we expect that for independent maps the methods in [Reference Aimino, Nicol and Vaienti2] will yield results like Theorems 2.15–2.17 as well. The novelty in Theorems 2.15–2.17 is that we show how to apply the method of cumulants in the context of skew products with nonindependent fiber maps, which results in concentration inequalities, moderatedeviations principles and Gaussian moment estimates beyond the annealed setup [Reference Aimino, Nicol and Vaienti2].
Finally, let us consider the random function ${\mathcal S}_n(t)=n^{1/2}(S_{[nt]}{\mathbb E}[S_{nt}])$ on $[0,1]$ . We also obtain a functional CLT.
Theorem 2.19. Let ${\mathcal L}$ be a good cocycle. Suppose that $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ and that Assumption 2.13 holds true. Then the random function ${\mathcal S}_n$ converges in distribution towards the distribution of $\{sW_t\}$ , where W is a standard Brownian motion (restricted to $[0,1]$ ) and $s^2$ is the asymptotic variance.
Remark 2.20. The proof of Theorem 2.19 appears in §3.3. In [Reference Aimino, Nicol and Vaienti2] an ASIP was obtained, which yields the functional CLT. In §2.5.2 below, using different mixing coefficients for the base map, we will obtain an ASIP for the more general skew products considered in this paper. However, Theorem 2.19 shows that the functional CLT already holds true for stretched exponential ${\alpha }$ mixing base maps.
2.5.2 An almost sure invariance principle and exponential concentration inequalities for $\phi $  and $\psi $ mixing driving processes (via martingale methods)
Let $({\Omega }_0,\mathscr F,\textbf {P})$ be the probability space on which $(\xi _n)$ is defined. We recall that the $\phi $ mixing and $\psi $ (dependence) coefficient between two sub ${\sigma }$ algebras ${\mathcal G},{\mathcal H}$ of $\mathscr F$ is given by
and
The reverse $\phi $ mixing coefficients of $(\xi _n)$ are defined by
while the $\psi $ mixing coefficients of $(\xi _n)$ are defined by
where $\mathscr F_{\infty ,k}$ is the $\sigma $ algebra generated by $\xi _j$ , $j\leq k$ , and $\mathscr F_{k+n,\infty }$ is generated by $\xi _j, j\geq k+n$ . It is clear from the definitions of the mixing coefficients that
Theorem 2.21. (Exponential concentration and maximal inequalities)
Let ${\mathcal L}$ be a good cocycle. Suppose the observable satisfies $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })^2\\varphi _{\omega }\_{BV})<\infty $ .
Let ${\mathcal F}_0$ be the $\sigma $ algebra generated by the map $\pi ({\omega },x)=(({\omega }_j)_{j\geq 0},x)$ , namely the one generated by ${\mathcal B}$ and the coordinates with nonnegative indexes in the ${\omega }$ direction. If either $\mathrm {essinf}\inf _{x} h_{\omega }(x)>0$ and $\sum _{n}\phi _{n,R}<\infty $ or $\sum _{n}\psi _{n}<\infty $ then there is an ${\mathcal F}_0$ measurable function $\chi \in L^\infty (\mu )$ such that if we set $u=\varphi +\chi \circ \tau \chi $ then $(u\circ \tau ^n)$ is a reverse martingale difference with respect to the reverse filtration $\{\tau ^{n}{\mathcal F}_0\}$ . As a consequence, we have the following assertions.

(i) There are constants $a_1,a_2,a_3>0$ such that the following exponential concentration inequality holds true: for every $t>0$ , we have
(2.25) $$ \begin{align} \mu(S_n{\mathbb E}_\mu[S_n]\geq tn+a_1)\leq a_2e^{a_3 nt^2}. \end{align} $$The constants $a_1,a_2,a_3$ depend only on $\tilde \Phi =\sum _{n}\phi _{n,R}<\infty $ and c (or $\tilde \Psi =\sum _{n}\psi _{n}<\infty $ ), the constant $C_{\mathrm {v}}$ and $\\varphi \_{K,2}=\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })^2\\varphi _{\omega }\_{BV})$ , and an explicit formula for them can be recovered from the proof. 
(ii) For every $p\geq 2$ , we have
(2.26) $$ \begin{align} \Big\\!\max_{1\leq k\leq n}S_k{\mathbb E}[S_k]\Big\_{L^p(\mu)}\leq C_pn^{1/2}, \end{align} $$where $C_p>0$ is a constant (which can be recovered from the proof and depends only on p and the above constants).
We refer readers to [Reference Korepanov and Leppanen43] for some related moment bounds for random intermittent maps.
The proof of Theorem 2.21 appears in §4. Let us note that once the martingalecoboundary representation $\varphi =u+\chi \chi \circ \tau $ is established, Theorem 2.21(i) follows from the Azuma–Hoeffding inequality together with Chernoff’s bounding method, and Theorem 2.21(ii) follows from the socalled Rio inequality [Reference Rio48] (see [Reference Merlevéde, Peligrad and Utev45, Proposition 7]).
To obtain the martingalecoboundary representation we show that if ${\mathcal K}$ is the transfer operator (namely, the one satisfying the duality relation
corresponding to the system $({\Omega }\times X,{\mathcal F}_0,\mu ,\tau )$ then there is a constant $C>0$ such that
where $\gamma _n$ is either $\psi _{n}$ or $\phi _{n,R}$ , depending on the case, and ${\delta }\in (0,1)$ . Once this is established we can take
The proof of (2.27) is given in Proposition 4.3 (i).
Our next result is an ASIP.
Theorem 2.22. (ASIP)
Let ${\mathcal L}$ be a good cocycle, and suppose that the observable satisfies $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })^2\\varphi _{\omega }\_{BV})<\infty $ .
When $\mathrm {essinf}\,\inf _{x} h_{\omega }(x)>0$ we set $\gamma _n=\phi _{R,n}$ , while otherwise we set $\gamma _n=\psi _n$ . In both cases, assume that
and
Then the limit
exists and the following version of the ASIP holds true: there is a coupling of $(\varphi \circ \tau ^n)$ with a sequence of i.i.d. Gaussian random variables $Z_j$ with zero mean and variance $s^2$ such that
Remark 2.23. The ASIP implies the functional CLT, see [Reference Philipp and Stout47]. Thus, Theorem 2.22 yields better results than Theorem 2.19 for $\phi _R$  or $\psi $ mixing driving sequences (which are not necessarily stretched exponentially mixing).
The proof of Theorem 2.22 appears in §4 and relies on an application of [Reference Cuny and Merlevede13, Theorem 3.2]. In addition to (2.27), in order to apply [Reference Cuny and Merlevede13, Theorem 3.2] we will show that for all $1\leq i,j\leq n$ we have
where $\bar \varphi =\varphi \mu (\varphi )$ , C is a constant and ${\delta }$ and $\gamma _n$ are as in (2.27). The proof of (2.28) is given in Proposition 4.3 (ii).
Remark 2.24. As discussed in §1.3.2, the martingalecoboundary decomposition in Theorem 2.21 (and its consequences) is comparable with the annealed case [Reference Aimino, Nicol and Vaienti2], and the main novelty is that we obtain it for more general skew products and functions $\varphi $ which depend on ${\omega }$ . Moreover, we do not assume that all $T_{\omega }$ preserve the same absolutely continuous probability measure. The ASIPs we obtain are comparable to ASIPs in [Reference Aimino, Nicol and Vaienti2] (see the discussion in §1.3.2).
2.5.3 A vectorvalued almost sure invariance principle in the uniformly random case for exponentially fast $\alpha $ mixing base maps
Let us take a vectorvalued measurable function $\varphi =(\varphi _1\ldots \varphi _d):{\Omega }\times X\to {\mathbb R}^d$ such that $\varphi _{\omega }=\varphi ({\omega },\cdot )$ depend on ${\omega }$ only through ${\omega }_0$ and $\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{{\omega },i}\_{BV})<\infty $ for all $1\leq i\leq d$ . Let us also assume that $\mu (\varphi _i)=0$ for every i. Set $S_n=\sum _{j=0}^{n1}\varphi \circ \tau ^j$ .
Theorem 2.25. Suppose that ${\alpha }_n=O({\alpha }^n)$ for some ${\alpha }\in (0,1)$ . Then there is a positive semidefinite matrix ${\Sigma }^2$ such that
Moreover, ${\Sigma }^2$ is positive definite if and only if $\varphi \cdot v \neq rr\circ \tau $ for all unit vectors v and all $r\in L^2$ .
Assume now that there are constants $C>0$ and ${\delta }\in (0,1)$ so that
namely, that $K({\omega })$ is a bounded random variable. Then there is a coupling of ${(\varphi \circ \tau ^n)}$ with a sequence of independent Gaussian centered random vectors $(Z_n)$ such that ${\mathrm {Cov}(Z_n)={\Sigma }^2}$ and for every ${\varepsilon }>0$ ,
3 Limit theorems via the method of cumulants for ${\alpha }$ mixing driving processes
We recall next that the kth cumulant of a random variable W with finite moments of all orders is given by
Note that ${\Gamma }_1(W)={\mathbb E}[W]$ , ${\Gamma }_2(W)=\mathrm {Var}(W)$ , and ${\Gamma }_k(aW)=a^k{\Gamma }_k(W)$ for any $a\in {\mathbb R}$ and $k\geq 1$ .
From now on we will assume that ${\mathbb E}[S_n]=0$ for all n, that is, we will replace $\varphi $ by $\varphi \mu (\varphi )$ . The main result in this section is the following theorem.
Theorem 3.1. Let ${\mathcal L}$ be a good cocycle, and suppose that Assumption 2.13 holds true and that $\\varphi \_{K}=\operatorname {\mathrm {esssup}}_{{\omega }\in {\Omega }}(K({\omega })\\varphi _{\omega }\_{BV})<\infty $ . Then, with $\gamma =1/\eta $ , there exists a constant $c_0$ which depends only on $\\varphi \_K$ and the constants from Assumption 2.13 such that, for any $k\geq 3$ ,
We will prove Theorem 3.1 by applying the following Proposition 3.3, which appears in [Reference Hafouta25] as Corollary 3.2.
Let us start with a few preparations. Let V be a finite set and $\rho :V\times V\to [0,\infty )$ be such that $\rho (v,v)=0$ and $\rho (u,v)=\rho (v,u)$ for all $u,v\in V$ . For every $A,B\subset V$ set
We assume here that there exist $c_0\geq 1$ and $u_0\geq 0$ such that
for all $v\in V$ and $s\geq 1$ .
Next, let $X_v,\, v\in V$ be a collection of centered random variables with finite moments of all orders, and for each $v\in V$ and $t\in (0,\infty ]$ let $\varrho _{v,t}\in (0,\infty ]$ be such that $\X_v\_t\leq \varrho _{v,t}$ .
Assumption 3.2. For some $0<{\delta }\leq \infty $ and all $k\geq 1$ , $b>0$ and a finite collection $A_j$ , $j\in {\mathcal J}$ , of (nonempty) subsets of V such that $\min _{i\not =j}\rho (A_i,A_j)\geq b$ and $r:=\sum _{j\in {\mathcal J}} A_j\leq k$ , we have
where ${\gamma }_{\delta }(b,r)$ is some nonnegative number which depends only on ${\delta },b$ and r, and ${\Delta }$ stands for the cardinality of a finite set ${\Delta }$ .
Set $W=\sum _{v\in V} X_v$ . In the course of the proof of Theorems 2.14–2.16 and 2.19 we will need the following general result.
Proposition 3.3. [Reference Hafouta25, Corollary 3.2]
Suppose that inequality (3.1) and Assumption 3.2 are in force. Assume also that
for some $a,\eta>0$ , $d\geq 1$ and all $k,m\geq 1$ . Then there exists a constant c which depends only on $c_0,a,u_0$ and $\eta $ such that, for every $k\geq 2$ ,
where for all $q>0$ ,
When the $X_v$ are bounded and (3.2) holds true with ${\delta }=\infty $ we can always take ${\varrho _{v,t}=\varrho _{v,\infty }}$ , $t>0$ , and then, for any $k\geq 2$ ,
When