arXiv subjects
Publications and source records attributed to Yeor Hafouta.
We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.
We extend the spectral method for proving limit theorems to random non-uniformly expanding dynamical systems. This yields the CLT and moderate deviations principles (MDP). We show that as the amount of non-uniformity decreases the CLT rates and the speed in the MDP become closer to the optimal ones. For smooth systems the rates are effective. Compared to recent progress on the subject [34] we are able to consider much more general maps, Gibbs measures and observables. However, our main results are new even in the setup of [34].
We prove local central limit theorems for partial sums of the form \newline $\,S_n=\sum_{j=0}^{n-1}f_j\circ T_{j-1}\circ\cdots\circ T_1\circ T_0$ where $f_j$ are uniformly Hölder functions and $T_j$ are expanding maps. Using a symbolic representation a similar result follows for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$. All of our results are already new when all maps are the same $T_j=T$ but observables $(f_j)$ are different. The current paper compliments [43] where Berry--Esseen theorems are obtained. An important step in the proof is developing an appropriate reduction theory in the sequential case.
We prove Berry-Esseen theorems and the almost sure invariance principle with rates for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j\circ T_{j-1}\circ\cdots\circ T_1\circ T_0$ where $f_j$ are functions with uniformly bounded ``variation" and $T_j$ is a sequence of expanding maps. Using symbolic representations similar result follow for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$. All of our results are already new for a single map $T_j=T$ and a sequence of different functions $(f_j)$.
The purpose of this paper is to provide a first class of explicit sufficient conditions for the central limit theorem and related results in the setup of non-uniformly (partially) expanding non iid random transformations, considered as stochastic processes together with some random Gibbs measure. More precisely, we prove a central limit theorem (CLT), an almost sure invariance principle, a moderate deviations principle, Berry-Esseen type estimates and a moderate local central limit theorem for random Birkhoff sums generated by a non-uniformly partially expanding dynamical systems $T_ω$ and a random Gibbs measure $μ_ω$ corresponding to a random potential $ϕ_ω$ with a sufficiently regular variation.
We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel's approach for establishing ASIP and the recent construction of the so-called adapted norms, which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of a scaling condition, answering a question of the first and third authors.
In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast $ϕ$-mixing. The obtained rate is of order $o(V_n^{\frac14+\del})$ for an arbitrary $\del>0$, where $V_n$ is the variance of the underlying partial sums $S_n$. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.
We obtain convergence rates (in the Levi-Prokhorove metric) in the functional central limit theorem (CLT) for partial sums $S_n=\sum_{j=1}^{n}ξ_{j,n}$ of triangular arrays $\{ξ_{1,n},ξ_{2,n},...,ξ_{n,n}\}$ satisfying some mixing and moment conditions (which are not necessarily uniform in $n$). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of $S_n$. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of $S_n$ is "not negligible" in comparison with the (new) number of summands.
In \cite{MPU} a functional CLT was obtained for triangular arrays satisfying the Lindeberg condition, that the sum of the individual variances is at most the same order as the variance of the underlying sum, and under the optimal mixing rats $\sum_{n}ρ(2^n)<\infty$, where $ρ(\cdot)$ are the $ρ$-mixing coefficients of the array. In this paper we will present alternative conditions which do not involve the assumption on the sum of variances, and instead we will assume certain maximal moment assumptions (which we can verify for $ϕ$-mixing arrays) and mixing rates of the form $\sum_nρ(e^{G(n)})<\infty$ where $G(n)$ grows sub-linearly fast in $n$ (e.g. $G(n)=n/\ln(\ln n)$). We will also discuss alternative conditions to the ones in the functional CLT for $α$-mixing triangular arrays which was obtained in \cite{MP}.
In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (contrary to \cite{ANV}) is that the random maps are not independent, they do not preserve the same measure and the observable $φ$ depends also on the base space. For stretched exponentially $\al$-mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $ϕ$ or $ψ$ mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty$ convergence of the iterates $\cK^n$ of a certain transfer operator $\cK$ with respect to a certain sub-$\sig$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.
We prove a Berry-Esseen theorem and Edgeworth expansions for partial sums of the form $S_N=\sum_{n=1}^{N}f_n(X_n,X_{n+1})$, where $\{X_n\}$ is a uniformly elliptic inhomogeneous Markov chain and $\{f_n\}$ is a sequence of uniformly bounded functions. The Berry-Esseen theorem holds without additional assumptions, while expansions of order $1$ hold when $\{f_n\}$ is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of $f_n$ is of order $O(n^{-\be})$ for some $\be\in(0,1/2)$. In this case it turns out that expansions of any order $r<\frac1{1-2\be}$ hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When $f_n$ are uniformly Lipschitz continuous we show that $S_N$ admits expansions of all orders. When $f_n$ are uniformly Hölder continuous with some exponent $\al\in(0,1)$, we show that $S_N$ admits expansions of all orders $r<\frac{1+\al}{1-\al}$. For Hölder continues functions with $\al<1$ our results are new also for uniformly elliptic homogeneous Markov chains and a single functional $f=f_n$. In fact, we show that the condition $r<\frac{1+\al}{1-\al}$ is optimal even in the homogeneous case.
We prove a local central limit theorem for "nonconventional" sums generated by some classes of sufficiently fast mixing sequences.
We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.
We extend the spectral approach of S. Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behavior of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.
We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the underlying partial sums modulo some $h\in \bbN$ and the uniform distribution could fail to be $o(σ_N^{1-r})$, where $σ_N$ is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order $r$ fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal.
The classical Eagleson's theorem states that if appropriately normalized Birkhoff sums generated by a measurable function and a probability preserving transformation converge in distribution, then they also converge in distribution with respect to any probability measure which is absolutely continuous with respect to the invariant one. In this short note we prove several versions of Eagleson's theorem for some classes of non-stationary stochastic processes which satisfy certain type of decay of correlations.
The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for \emph{vector-valued} observables. We stress that the previous works considered exclusively the case of scalar-valued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviation principle, concentration inequalities, Berry-Esseen estimates as well as Edgeworth and large deviation expansions. Although our techniques rely on the approach developed in the previous works of the first author \textit{et al}., we emphasize that our arguments require several nontrivial adjustments as well as new ideas.
We prove a quenched almost sure invariance principle for certain classes of random distance expanding dynamical systems which do not necessarily exhibit uniform decay of correlations.