arXiv subjects

Yeor Hafouta

Publications and source records attributed to Yeor Hafouta.

At least 19 recordsLinked to original sources

arXiv2026

Local limit theorem and Edgeworth expansions for inhomogeneous random walks on $GL(d,\mathbb R)$

We prove a non-lattice local central limit theorem and Edgeworth expansions for the logarithm of the norms of products of invertible independent random matrices. Our conditions include a contraction assumption, an assumption that supports of the matrices are ``large enough" and their distributions are sufficiently regular. As a byproduct of the proof we are also able to provide a different proof to the optimal rates in the CLT proved in \cite{MatBE}. Like in \cite{MatBE} we provide several sufficient conditions for contraction.

math.PR↗

arXiv2026

Quenched and annealed linear response for some partially hyperbolic skew products

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products, with random hyperbolic maps on the fibers. The main novelty is that the base maps also depend on the parameter, which leads to different formulas in the linear response and the derivative of the asymptotic moments with respect to the parameter. Our annealed results apply to partially hyperbolic maps that are not covered in \cite{BashCastro26,Dol,DS}.

math.DS↗

arXiv2026

Berry-Esseen Bounds for Statistics of Non-Stationary, $\phi$-Mixing Random Variables

Using a modification of Stein's method, we generalize the results of Bentkus, G{\"o}tze, and Tikhomirov \cite{bentkus1997berry} to obtain Berry-Esseen bounds for a broad class of statistics of sequences of $\phi$-mixing, non-stationary random variables with polynomial mixing rates. %and linear variance. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary $\phi$-mixing random variables, including rates for a general class of processes of $\phi$-mixing random variables satisfying an aggregate third moment bound.

math.PR↗

arXiv2026

Effective geometric ergodicty for Markov chains in random environment

In this short note we prove ``effective" geometric ergodicity (i.e a Perron-Frobenius theorem) for Markov chains in random mixing dynamical environment satisfying a random non-uniform version of the Doeblin condition. Effectivity here means that all the random variables involved in the random exponential rates are integrable with arbitrarily large order. This compliments \cite[Theorem 2.1]{Kifer 1996}, where ``non-effective" geometric ergodicity was obtained. From a different perspective, our result is also motivated by egrodic theory, as it can be seen as an effective version of the ``spectral" gap in the top Oseledets space in the Oseledets multiplicative ergodic theorem for the random Markov operator cocycle (when it applies). We also present applications of the effective ergodicity to rates in the (quenched) almost sure invariance principle (ASIP), exponential decay of correlations for Markovian skew products and for exponential tails for random mixing times. As a byproduct of the proof of the ASIP rates we also provide easy to verify sufficient conditions for the verification of the assumptions of \cite[Theorem 2.4]{Kifer 1998}.

math.PR↗

arXiv2025

Limit theorems for inhomogeneous random walks on $GL(d,\mathbb R)$

We prove Berry-Esseen theorems, almost sure invariance principle rates and large deviations for products of independent but not identically distributed invertible matrices with some average (logarithmic) projective contraction and uniform boundedness assumptions. We also characterize the divergence of the variance of the logarithm of the norm of the product. Our approach is based on verifying the conditions of \cite{NewBE} after reversing time.

math.PR↗

arXiv2025

Non-uniform Edgeworth expansions for weakly dependent random variables and their applications

We obtain non-uniform Edgeworth expansions for several classes of weakly dependent (non-stationary) sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics. To the best of our knowledge this is the first time such results are obtained beyond the case of independent summands, even for stationary sequences. As an application of the non uniform expansions we obtain average versions of Edgeworth exapnsions, which provide estimates of the underlying distribution function in $L^p(dx)$ by the standard normal distribution function and its higher order corrections. An additional application is to expansions of expectations $\bbE[h(S_n)]$ of functions $h$ of the underlying sequence $S_n$, whose derivatives grow at most polynomially fast. In particular we provide expansions of the moments of $S_n$ by means the variance of $S_n$. A third application is to Edgeworth expansions in the Wasserstein distance (transport distance). In particular we prove Berry-Esseen theorems in the Wasserstein metrics. This paper compliments \cite{NonU BE} where non-uniform Berry-Esseen theorems were obtained.

math.PR↗

arXiv2025

Decay of correlations and limit theorems for random intermittent maps

In this paper, we revisit the problem of polynomial memory loss and the central limit theorem for time-dependent LSV maps. More precisely, we show that for random LSV maps corresponding to a random parameter beta() we obtain quenched memory loss, decay of correlations, central limit theorems with rates, moment bounds and almost sure invariance principles (ASIP) when the essential infimum of beta() is less than 1/5 and the driving process (i.e. random environment) is mixing sufficiently fast. In [59, Corollary 3.8] the ASIP was obtained for ergodic driving systems when the essential supremum of \b{eta} is less than 1/2. As will be elaborated in Section 1, restrictions on the essential infimum are more natural in our context. Our results have an abstract form which we believe could be useful in other circumstances, as will be elaborated in a future work.

math.DS↗

arXiv2025

Statistical properties of Markov shifts: part II-LLT

We prove Local Central Limit Theorems (LLT) for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is a Markov chains with equicontinuous conditional probabilities satisfying contraction conditions close in spirit to Dobrushin's, and some ``physicality" assumptions and $f_j$ are equicontinuous functions. Our conditions will always be in force when the chain takes values on a metric space and have uniformly bounded away from $0$ backward transition densities with respect to a measure which assigns uniform positive mass to certain ``balls". This paper complements \cite{MarShif1} where Berry-Esseen theorems, were proven for (not necessarily continuous) functions satisfying certain approximation conditions. Our results address a question posed by D. Dolgopyat and O. Sarig in \cite[Section 1.5]{DS}.

math.PR↗

arXiv2025

Limit theorems for inhomogeneous $\phi$-mixing Markov chains

We prove limit theorems for inhomogeneous $\phi$-mixing Markov chains. In the case of uniform $\phi$-mixing, we will focus on the purely sequential (inhomogeneous) setting, while in the non-uniform case we will focus on Markov chains in random dynamical environments. A large part of the paper is devoted to verifying our main results for Markov chains satisfying a lower Doeblin condition and for Markov chains satisfying a Dobrushin-type contraction condition. We then apply our results in the random environment case to prove estimates on random mixing times as well as limit theorems for Markovian skew products. In fact, we show that one can estimate the $\alpha$-mixing coefficients of the coordinates of the skew product, which opens the door to a variety of limit theorems. Compared with \cite{dolgopyat2023berry}, we are able to prove optimal CLT rates for inhomogeneous Markov chains solely under $\phi$-mixing, without any ellipticity assumptions. Compared with existing results for Markov chains in random ergodic environments, under mixing conditions on the environment, we are able to treat non-uniformly $\phi$-mixing chains, including non-uniform Doeblin/Dobrushin conditions, while all existing results either hold for uniformly mixing chains or are formulated under hard-to-verify conditions.

math.PR↗

arXiv2025

Statistical properties of Markov shifts (part I)

We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is an inhomogeneous Markov chain satisfying some mixing assumptions and $f_j$ is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions $f_j$ is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when $(X_j)$ are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}.

math.PR↗

arXiv2025

A few notes on the asymptotic behavior of Rademacher random multiplicative functions

Let $X_p, p\in\cP$ be a sequence of independent random variables s.t. $\bbP(X_p=\pm 1)=1/2$. Let $\te_j=\prod_{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. The from this point of view proving limit theorems for $S_n$ is natural problem, since $S_n$ mimics the behavior of $e^{\sqrt{\ln(\beta)}}$. It is a natural guiding conjecture that $S_n/\sqrt n$ obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite{[25]}) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite{[17]}, and by now it is a direct consequence of a more recent breakthrough by Harper \cite{Har20} that $\frac{S_n}{b_n}\to 0$ in $L^1$, where $b_n=(n^{1/2}(\ln(\ln(n)))^{-1/4})u_n, u_n\to\infty$. In particular $S_n/\sqrt n\to 0$. Nevertheless, the question whether there exists a sequence $a_n=o(b_n)$ such that $S_n/a_n$ converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by \cite{Gor1}. In this paper make an attempt to shed some light on the convergence of $S_n/a_n$. Additionally, we obtain explicit estimates on hight moments of $S_n$ without restrictions on the size of the moment compared to $n$ like in \cite[Theorem 1.2]{Har19}, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for $S_n$ (in the large deviations regime)

math.PR↗

arXiv2025

Liv\v{s}ic regularity for random and sequential dynamics through transfer operators

We prove Liv\v{s}ic-type regularity results of coboundary representations for non-autonomous dynamical systems. Our results have an abstract nature and apply to several important specific situations, such as (higher-dimensional) random or sequential piecewise expanding maps and subshifts of finite type, which have applications to Markov interval maps and to finite state inhomogeneous elliptic Markov shifts, via symbolic representations. We also obtain results for some classes of non-autonomous hyperbolic systems. Our results can be seen as non-autonomous versions of a recent result obtained by Morris. However, we emphasize that our proof differs from the one mentioned previously even in the deterministic case. Finally, we show that our results provide a more relaxed characterization for having variance growth of Birkhoff sums on random and sequential dynamical systems; we show that such growth can fail only when the underlying functions are a coboundary without special restrictions on the regularity of the coboundary. For random systems, we show that this is equivalent to having a coboundary with bounded ``variation", but for sequential systems it turns out that this is no longer true, as demonstrated by examples.

math.DS↗

arXiv2024

Local limit theorems for expanding maps

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 H\"older 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 H\"older 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.

math.DS↗

arXiv2024

Effective quenched linear response for random dynamical systems

We prove ``effective'' linear response for certain classes of non-uniformly expanding random dynamical systems which are not necessarily composed in an i.i.d manner. In applications, the results are obtained for base maps with a sufficient amount of mixing. The fact that the rates are effective is then applied to obtain the differentiability of the variance in the CLT as a function of the parameter, as well as the annealed linear response. These two applications are beyond the reach of the linear response obtained in the general case, when all the random variables appearing in the bounds are only tempered. We also provide several wide examples of one-dimensional maps satisfying our conditions, as well as some higher-dimensional examples.

math.DS↗

arXiv2024

Rates of convergence in CLT and ASIP for sequences of expanding maps

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 $(f_j)$.

math.DS↗

arXiv2023

Iterated invariance principle for random dynamical systems

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.

math.DS↗

arXiv2023

Spectral methods for limit theorems for random expanding transformations

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].

math.DS↗

arXiv2023

Explicit conditions for the CLT and related results for non-uniformly partially expanding random dynamical systems via effective RPF rates

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.

math.DS↗