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

1.2.1 Annealed limit theorems: i.i.d. maps

A particular well-studied 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 so-called annealed transfer operator, given by (see [Reference Baladi8, Reference Baladi and Young9, Reference Ishitani35])

$$ \begin{align*} {\mathcal A} g(x)=\int {\mathcal L}_{\omega} g(x)\,d{\mathbb P}({\omega}), \end{align*} $$

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 large-deviations principle and an ASIP) for random variables of the form

$$ \begin{align*} S_n\varphi({\omega},x)=\sum_{j=0}^{n-1}\varphi(T_{{\omega}_{j-1}}\circ\cdots\circ T_{{\omega}_0}x), \end{align*} $$

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

(1.1) $$ \begin{align} {\mathcal A}^n g=\int {\mathcal L}_{{\omega}}^ng\,d{\mathbb P}({\omega}), \end{align} $$

where ${\mathcal L}_{\omega }^n={\mathcal L}_{{\sigma }^{n-1}{\omega }}\circ \cdots \circ {\mathcal L}_{{\sigma }{\omega }}\circ {\mathcal L}_{\omega }$ , 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: non-i.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 non-independent 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 non-uniform 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 non-independent 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 non-independent 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.

1.3.1 Limit theorems for skew products: (functional) CLT, moment estimates, moderate-deviations 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

$$ \begin{align*} {\alpha}({\mathcal G},{\mathcal H})=\sup\{|\textbf{P}(A\cap B)-\textbf{P}(A)\textbf{P}(B)|: A\in{\mathcal G}, B\in{\mathcal H}\}. \end{align*} $$

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

(1.2) $$ \begin{align} {\alpha}_n=\sup_{k}{\alpha}({\mathcal F}_{-\infty,k}, {\mathcal F}_{k+n,\infty})={\alpha}({\mathcal F}_{-\infty,0},{\mathcal F}_{n,\infty}) \end{align} $$

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

$$ \begin{align*} s^2=\lim_{n\to\infty}\frac 1n\mathrm{Var}_\mu(S_n),\quad S_n=S_n\varphi, \end{align*} $$

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

$$ \begin{align*} \sup_{t\in{\mathbb R}}\bigg|\mu(S_n\leq ts\sqrt n)-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^t e^{-(1/2)x^2}dx\bigg|\leq Cn^{-1/({2+4\gamma})},\quad \gamma=1/\eta. \end{align*} $$

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 large-deviations results, often referred to as a moderate-deviations principle (see [Reference Dembo and Zeitouni14]). These results yield, for instance, that for every closed interval $[a,b]$ we have

$$ \begin{align*} \lim_{n\to\infty}\frac{1}{a_n^2}\ln\mu\bigg\{({\omega},x):\frac{S_n({\omega},x)}{a_nsn^{1/2}}\in[a,b]\bigg\}=-\frac12\inf_{x\in[a,b]}x^2, \end{align*} $$

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)^{k-2}$ 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, large-deviations 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 Rosenthal-type 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 so-called martingale-coboundary 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, González-Tokman 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

$$ \begin{align*} \phi({\mathcal G},{\mathcal H})=\sup\{|{\mathbb P}(B|A)-{\mathbb P}(B)|: A\in{\mathcal G}, B\in{\mathcal H}, {\mathbb P}(A)>0\} \end{align*} $$

and

$$ \begin{align*} \psi({\mathcal G},{\mathcal H})=\sup\bigg\{\bigg|\frac{{\mathbb P}(A\cap B)}{{\mathbb P}(A){\mathbb P}(B)}-1\bigg|: A\in{\mathcal G}, B\in{\mathcal H}, {\mathbb P}(A){\mathbb P}(B)>0\bigg\}. \end{align*} $$

The reverse $\phi $ -mixing coefficients of the sequence of coordinates $({\omega }_n)$ are defined by

(1.3) $$ \begin{align} \phi_{n,R}=\sup_{k}\phi({\mathcal F}_{k+n,\infty},{\mathcal F}_{-\infty,k})=\phi({\mathcal F}_{n,\infty},{\mathcal F}_{-\infty,0}), \end{align} $$

while the $\psi $ -mixing coefficients of $(\xi _n)$ are defined by

(1.4) $$ \begin{align} \psi_{n}=\sup_{k}\psi({\mathcal F}_{-\infty,k},{\mathcal F}_{k+n,\infty})=\psi({\mathcal F}_{-\infty,0},{\mathcal F}_{n,\infty}), \end{align} $$

where ${\mathcal F}_{n,m}$ is as defined before (1.2). It is clear from the definitions of the mixing coefficients that

$$ \begin{align*} {\alpha}_n\leq \phi_{n,R}\leq \psi_n. \end{align*} $$

When the sequence $({\omega }_n)$ is (sufficiently fast) $\phi $ - or $\psi $ -mixing we obtain a certain type of $L^\infty $ martingale-coboundary 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 $ martingale-coboundary 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

$$ \begin{align*} {\mathbb P}(|S_n-{\mathbb E}[S_n]|\geq tn+c_1)\leq c_2e^{-c_3 nt^2},\quad t>0, \end{align*} $$

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

$$ \begin{align*} \Big\|\!\max_{1\leq k\leq n}|S_k-{\mathbb E}[S_k]|\Big\|_{L_p}=O(n^{1/2}) \end{align*} $$

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 martingale-coboundary 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

$$ \begin{align*} \int ({\mathcal K} g) f\,d\mu=\int ({\mathcal K} g\circ\tau)f\circ\tau \,d\mu=\int {\mathbb E}[g|\tau^{-1}{\mathcal F}_0]\cdot f\circ\tau \,d\mu= \int g\cdot f\circ\tau \,d\mu \end{align*} $$

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

(1.5) $$ \begin{align} \|{\mathcal K}^{\,n}\varphi-\mu(\varphi)\|_{L^\infty}\leq C\|\varphi\|_{K,2}\cdot\gamma_n, \end{align} $$

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 martingale-coboundary representation is the ASIP, which in our context concerns almost sure approximation of the Birkhoff sum by Gaussians. The ASIP for random (and sequential) dynamical systems has been studied by several authors in recent years (see, for instance, [Reference Dragičević, Froyland, González-Tokman 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 $W_n=\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ . 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

$$ \begin{align*} \sup_{1\leq k\leq n}\bigg|W_k-\sum_{j=1}^k Z_j\bigg|=O(n^{1/4}(\log n)^{1/2}(\log\log n)^{1/4})\quad\text{almost surely}. \end{align*} $$

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

$$ \begin{align*} \|{\mathcal K}^i(\bar\varphi {\mathcal K}^j\bar\varphi)-\mu({\mathcal K}^i(\bar\varphi {\mathcal K}^j\bar\varphi))\|_{L^\infty} \end{align*} $$

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 martingale-coboundary approach. Gouëzel’s approach was also used in [Reference Atnip5] to obtain an ASIP for non-independent 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 vector-valued 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 non-conventional 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]).