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Yeor Hafouta

Publications and source records attributed to Yeor Hafouta.

46 records · Page 3Linked to original sources
arXiv2020

A vector valued almost sure invariance principle for time dependent non-uniformly expanding dynamical systems

We prove a vector-valued almost sure invariance principle for some classes of time dependent non-uniformly distance expanding dynamical systems. The models we have in mind are certain sequential versions of the smooth non-uniformly distance expanding maps considered in \cite{castro} and \cite{Vara}, as well as certain types of sequences of covering maps. Our results rely on the theory of complex projective metrics which was developed in \cite{Rug}, together with the spectral methods of Gouëzel \cite{GO}. A big advantage in applying the theory of complex cones here is that it also yields additional probabilistic limit theorems for random dynamical systems, as described at the last section of this paper.

math.DS↗
arXiv2020

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

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

math.DS↗
arXiv2020

Complex projective metrics on Young towers, countable sub-shifts and other countable covering maps

In this article we will apply complex projective metrics to sequences of complex transfer operators generated by Young towers, countable shifts and other types of distance expanding maps (possibly time dependent) with countable degrees. We will derive a sequential Ruelle-Perron-Frobenius theorem for sequence of complex operators defined by such maps, and relying on it a variety of probabilistic limit theorems (finer than the CLT) for non-stationary processes follows.

math.DS↗
arXiv2019

Limit theorems for some time dependent expanding dynamical systems

In this paper we will prove various probabilistic limit theorems for some classes of distance expanding sequential dynamical systems (SDS). Our starting point here is certain sequential complex Ruelle-Perron-Frobenius (RPF) theorems which were proved in \cite{book} and \cite{ASIP me} using contraction properties of a complex version of the projective Hilbert metric developed in \cite{Rug}. We will start from the growth rate of the variances of the underlying partial sums. This is well understood in the random dynamics setup, when the maps are stationary, but not in the SDS setup, where various growth rates can occur. Some of our results in this direction rely on certain type of stability in these RPF theorems, which is one of the novelties of this paper. Then we will provide general conditions for several classical limit theorems to hold true in the sequential setup. Some of our general results mostly have applications for composition of random non-stationary map, while the conditions of the other results hold true for general type of SDS. In the latter setup, results such as the Berry-Esseén theorem and the local central limit theorem were not obtained so far even for independent but not identically distributed maps, which is a particular case of the setup considered in this paper.

math.DS↗
arXiv2018

Limit theorems for some skew products with mixing base maps

We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\om,x)=(\te\om,T_\om x)$ together with a $T$-invariant measure, whose base map $\te$ satisfies certain topological and mixing conditions and the maps $T_\om$ on the fibers are certain non-singular distance expanding maps. Our results hold true when $\te$ is either a sufficiently fast mixing Markov shift or a (non-uniform) Young tower with at least one periodic point and polynomial tails. %In fact, our conditions will be satisfied %when $\om$ is the whole orbit the towers. The proofs are based on the random complex Ruelle-Perron-Frobenius theorem from \cite{book} applied with appropriate random transfer operators generated by $T_\om$, together with certain regularity assumptions (as functions of $\om$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition will also be obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in \cite{Aimino} does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.

math.PR↗
arXiv2018

Stein's method for nonconventional sums

We obtain almost optimal convergence rate in the central limit theorem for "nonconevntional" sums of the form $S_N=N^{-\frac12}\sum_{n=1}^N (F(ξ_n,ξ_{2n},...,ξ_{\ell n})-\bar F)$.

math.PR↗