Repetition times for Gibbsian sources

In this paper we consider the class of stochastic stationary sources induced by one-dimensional Gibbs states, with Holder continuous potentials. We show that the time elapsed before the source repeats its first n symbols, when suitably renormalized, converges in law either to a log-normal distribution or to a finite mixture of exponential random variables. In the first case we also prove a large deviation result.

Automatic fitting of cochlear implants with evolutionary algorithms

This paper presents an optimisation algorithm designed to perform in-situ automatic fitting of cochlear implants.All patients are different, which means that cochlear parametrisation is a difficult and long task, with results ranging from perfect blind speech recognition to patients who cannot make anything out of their implant and just turn it off.The proposed method combines evolutionary algorithms and medical expertise to achieve autonomous interactive fitting through a Personal Digital Assistant (PDA).

The pianigiani-yorke measure for topological markov chains

We prove the existence of a Pianigiani-Yorke measure for a Markovian factor of a topological Markov chain. This measure induces a Gibbs measure in the limit set. The proof uses the contraction properties of the Ruelle-Perron-Frobenius operator.

Interactive evolution for cochlear implants fitting

International audience; Cochlear implants are devices that become more and more sophisticated and adapted to the need of patients, but at the same time they become more and more difficult to parameterize. After a deaf patient has been surgically implanted, a specialised medical practitioner has to spend hours during months to precisely fit the implant to the patient. This process is a complex one implying two intertwined tasks: the practitioner has to tune the parameters of the device (optimisation) while the patient's brain needs to adapt to the new data he receives (learning). This paper presents a study that intends to make the implant more adaptable to environment (auditive ecology) and…

Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems

In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functio…

Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems

In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class…

A New Universal Cellular Automaton Discovered by Evolutionary Algorithms

In Twenty Problems in the Theory of Cellular Automata, Stephen Wolfram asks “how common computational universality and undecidability [are] in cellular automata.” This papers provides elements of answer, as it describes how another universal cellular automaton than the Game of Life (Life) was sought and found using evolutionary algorithms. This paper includes a demonstration that consists in showing that the presented R automaton can both implement any logic circuit (logic universality) and a simulation of Life (universality in the Turing sense).

Quasi-Stationary Distribution and Gibbs Measure of Expanding Systems

Let T be an expanding transformation defined on A —(J A{, i= 1being a finite collection of connected open bounded subsets of 2Rn,such that T Acontains strictly Aand Tis Markovian. We prove the existence of a quasi-stationary distrition for T. We show that the T-invariant probability on the limit Cantor set is Gibbsian with potential Log|_DT|. Using the Hilbert projective metric we prove that both distributions are weak limits of conditional laws of probabilities, the speed of convergence being exponential. These results develop a previous work by G. Pianigiani and J.A. Yorke.

Using interactive evolutionary algorithms to help fit cochlear implants

On the enhancement of diffusion by chaos, escape rates and stochastic instability

We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.

On the existence of conditionally invariant probability measures in dynamical systems

Let T : X→X be a measurable map defined on a Polish space X and let Y be a non-trivial subset of X. We give conditions ensuring the existence of conditionally invariant probability measures to non-absorption in Y. For dynamics which are non-singular with respect to some fixed probability measure we supply sufficient conditions for the existence of absolutely continuous conditionally invariant measures. These conditions are satisfied for a wide class of dynamical systems including systems that are Φ-mixing and Gibbs.