Monday June 11

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List of talks

Jon Aaronson (Tel Aviv university, Israel) Entropy of conservative endomorphisms We discuss entropy (as defined by Krengel in 1967) of infinite, conservative transformations introducing a similarity-invariant class of quasi-finite transformations (which means that the entropy of 1st return time partition is finite for some set). For these transformations there is a Pinsker - (i.e. maximal zero entropy-) factor and information convergence. In certain nice cases, we obtain distributional convergence of information. It turns out that there are probability preserving transformations with zero entropy with analogous properties. Joint work with Kyewon Koh Park.

Richard Bradley (Indiana university, USA) A strictly stationary, 5-tuplewise independent counterexample to the central limit theorem A strictly stationary random sequence (Xk, kZ) is constructed with the following properties: The random variables Xk each take just the values −1 and +1, with probability 1/2 each; every five of the random variables Xk are independent; and for every infinite set Q of positive integers, there exists an infinite set TQ and a nondegenerate, nonnormal probability measure μ on the real line (μ may depend on Q) such that Sn/√n converges in distribution to μ as n → ∞, nT. (Here Sn := X1 + X2 + ... + Xn.) This example complements the strictly stationary, pairwise independent counterexamples (to the central limit theorem) constructed by Janson [Stochastics 23 (1988) 439-448]; the strictly stationary, three-state, absolutely regular, triplewise independent counterexample developed in two papers by the author [Probab. Th. Rel. Fields 81 (1989) 1-10, Rocky Mountain J. Math. (in press)]; and also the N-tuplewise independent, identically distributed (but not strictly stationary) counterexamples constructed by Pruss [Probab. Th. Rel. Fields 111 (1998) 323-332] for arbitrary positive integers N.

Youri Davydov (Lille 1) On a non-classical invariance principle At the beginning of the talk a general discussion of the notion of asymptotical proximity of two sequences of probability measures is given.
After that we consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, we explore the difference of the following two distributions in the space C (of continuous functions on [0,1]). The first is the distribution of the continuous piecewise linear partial-sum process generated by a sequence of independent random variables, and the second is the distribution of the similar process generated by the sequence of normal r.v.'s with the same first two moments. The novelty is that the condition of negligibility of the r.v.'s is not imposed. We establish a necessary and sufficient condition of the weak convergence of the difference mentioned to zero measure in C.
(Avec V. Rotar.)
Yu. Davydov, V. Rotar, On a non-classical invariance principle, (2007) preprint, ArXiv, math.PR/0702085

Herold Dehling (Ruhr University, Bochum, Germany) A law of large numbers for rescaled random difference equations We study the behavior of stochastic processes defined as an iterated function system Xn+1=Xn+af(Xn,Un+1) with initial value X0=x0 and a stationary ergodic input signal (Un)n≥ 1 for small values of the parameter a. We prove almost sure convergence of the sample path to the solution of the ordinary differential equation y'=F(y), where F(y)=E(f(y,U)). The results have applications in the study of neural network learning algorithms.

Yves Derriennic (Brest) Central limit theorem for random walks on orbits of transformations Given an ergodic measure-preserving automorphism τ of a probability space (S, S, m) and a probability measure ν on the integer line Z, we consider the Markov chain (Xn) built on the Markov operator P = Σk ν(kk . We give conditions on functions fL2(m) insuring the validity m-almost surely of the central limit theorem for sums Σ0n f(Xk) with respect to the Markovian law with starting point x. Special emphasis is put on the case where the probability measure ν is not centered.

Paul Doukhan (ENSAE Paris) Weak dependence models and some applications This talk is aimed at providing some features of weak dependence conditions introduced in Doukhan and Louhichi (1999). The standard models used in statistics satisfy such properties and several new models of random processes and random fields are also described. After a rapid list applications of weak dependence, we detail some applications of a Lindeberg lemma for dependent sequences: subsampled partial sums, density estimates and subsampled density estimates are direct applications of this result while the standard central limit theorem asks more work.

Olivier Durieu (Rouen) Contre-exemple autour du principe d'invariance faible Le but de cet exposé est de comparer trois critères conduisant au théoreme limite centrale et au principe d'invariance faible : la décomposition martingale-cobord, un critère projectif introduit par Dedecker et Rio et la condition de Maxwell et Woodroofe. On montre que dans tout système dynamique ergodique, ces trois critères sont indépendants.

El Houcein El Abdalaoui (Rouen) On the singularity of rank one transformations and CLT We introduce a new tool to study the spectral type of rank-one transformations using the method of central limit theorem for trigonometric sums. We get a new class of rank-one maps with singular spectrum and a simple proof of Bourgain theorem on the singularity of Ornstein class of rank-one maps.

Katusi Fukuyama (Kobe University, Japan) The law of the iterated logarithm for discrepancies of θnx It is known that the sequence {θk x} is uniformly distributed mod 1 for almost every x. The speed of convergence to uniform distribution is measured by discrepacies DN, and we show the exact law of the iterated logarithm

N→ ∞
= Σθ
By investigating the LIL of Riesz-Raikov sums, we have succeeded in determing Σθ explicitly for wide class of θ.

For example, Σθ= 1/2, if θ is not a root of rational number.

Let θ = (p/q)1/r with p, qN, r=min{kN∣ θkQ}, and gcd(p,q)=1. If p and q are odd numbers, Σθ= √(pq+1)/(pq−1)/2. Especially, when p is odd and q=1, Σθ= √(p+1)/(p−1)/2. If p≥ 4 is even and q=1,
Σθ= √ (p+1)p(p−2)/(p−1)3/2. If p=2 and q=1, Σθ= √42/9.

Mikhail Gordin (St. Petersburg, Russia) V-statistics of a measure preserving transformation: CLT for canonical case Let T be an ergodic measure preserving transformation of probability space (X,F, P). For a suitable symmetric function h: XmR the expression

Vn(h)(x) = n m
1 ≤ i1, ..., imn
h(Ti1x, ..., Timx), xX,
defines a sequence (Vn(h))nm of functions on X called V−statistics of order m (with the kernel h). In this formula we need to consider h restricted to some subsets of Xm which have the product measure zero. This does not make sense for a general measurable h or for hL2(Xm,Fm, Pm), and we need to work within classes of more regular functions.

We consider the asymptotic behavior of V−statistics in the setting when a T−invariant filtration is specified. For simplicity we restrict ourself to the case of a non-invertible exact T which presents a suitable setup for applying reversed martingale approximation and similar methods. However, this setup requires rather restrictive assumptions about the kernel h. In particular, for m=2 we deal with trace class kernels only. An equivalent assumption can be stated for general m ≥ 2 in terms of the projective tensor power of L2(X, F, P). A kind of the Hoeffding decomposition was introduced and a version of the Central Limit Theorem (CLT) for so-called non-degenerate case was proved in a previous paper. In the talk we deal with so-called totally degenerate (or canonical) V-statistics and prove a version of the CLT in this setting. The talk is based on a joint work with H. Dehling and M. Denker.

Sébastien Gouëzel (Rennes 1) Théorème de la limite locale pour des applications non-uniformement partiellement hyperboliques, et suite de Farey On considère des produits gauches de la forme (x,ω)↦ (Tx, ω+ϕ(x)) où T est une application non-uniformément dilatante sur un espace X, préservant une mesure μ (eventuellement singuliere par rapport à la mesure de Lebesgue), et ϕ est une fonction C1 de X dans le cercle. Sous des hypothèses assez faibles sur μ et ϕ, on démontre qu'une telle application mélange exponentiellement et satisfait le théorème de la limite locale. Ces résultats s'appliquent à une marche aléatoire de nature arithmétique, reliée à la suite de Farey.

Yonatan Gutman (The Hebrew University of Jerusalem, Israel) On processes which cannot be distinguished by finite observations A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions sn such that sn(x1...xn) → J(X) in probability for every process X=(xn) belonging to C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy. We sharpen this in several ways. Our main result is that if XY is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X or Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss' result, and extends it to many other families of processes, e.g. it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure.
Joint work with Michael Hochman.
See also arXiv:math.DS/0608310.

Loic Hervé (INSA Rennes) Limit theorems for stongly ergodic Markov chains Let (Xn)n be a Markov chain, with state space E, invariant distribution ν, transition probability Q, and satisfying the following strong ergodicity condition : the iterates of Q converge to ν in operator norm w.r.t to a certain Banach space. Let ξ : ER. The purpose of this talk is to show how the perturbation theorem of Keller-Liverani and martingale methods enable to improve the spectral method in order to establish for (ξ(Xn))n some limit theorems (such as Local Limit Th., Berry-Esseen type Th., Renewal Th.). In particular, in the context of geometrical ergodic Markov chains and iterative models, this new approach provides these limit theorems under the expected moment conditions on ξ.

Hiroshi Ishitani (Mie University, Japan) Invariant measures for a class of transformations on a real line and their ergodic properties A various kind of 1-dimensional transformations have been found to have absolutely continuous invariant measures. However, there are not many transformations whose densities are explicitly known. This talk is concerned with giving explicitly the invariant density for some class of transformations from the real line into itself. Using Cauchy's integral formula, we show that the invariant density can be written in terms of the complex fixed point or in terms of the complex periodic point with period 2. The explicit form of the invariant density allows us to obtain the ergodic properties of the transformation by using the known results for the transformations on the finite interval.

Joanna Jaroszewska (Warsaw, Poland) And yet it converges During the talk I would like to provide an answer to the question 1 posed by A. Johansson and A. Oberg in their recent preprint "Square summability of variations and the transfer operator". The question concerns the limit behaviour of iterates of the Ruelle's operator defined by continuous g-measures.

Brunon Kaminski (Torun, Poland) Density of the measure-theoretic directional entropy It is introduced the concept of a density for the directional entropy of a lattice dynamical system. This concept is a measure-theoretic analogue of the topological density considered by Afraimovich, Courbage, Fernandez and Morante.
One proves that if the considered measure-preserving transformation T commutes with the shift transformation S, then the density is a constant function and equals the Conze-Katznelson-Weiss entropy of a Z2- action generated by T and S. Considering the product transformation one shows that in general the density may be not constant.

Mike Keane (Wesleyan University, USA) A Bernoulli shift in the M/M/1 queue The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this lecture we show that the associated measure-preserving transformation is isomorphic to a Bernoulli shift. If time permits, we discuss some extensions of Burke's theorem where it remains open to determine if, or when, the analogue of this result holds. This is joint work with Neil O'Connell.

Bryna Kra (Northwestern University, USA) Nilsystems in Ergodic Theory Nilsystems play an important role in recent developments in combinatorial ergodic theory. I will give an overview of their use, with a focus on the interactions with additive combinatorics and harmonic analysis.

Thomas Langlet (Université du Littoral) Continuité de séries de fonctions le long de marches aléatoires On s'intéresse à la convergence de séries aléatoires définies par le produit de nombres complexes et une fonction 1-périodique appliquée à une marche aléatoire. On est amené à différencier trois cas :

On obtient une convergence uniforme dans le cas continu et une convergence presque partout dans les deux autres cas.

François Ledrappier (University of Notre-Dame, USA) Propriété de Liouville et mouvement brownien pour des revêtements Pour un revêtement régulier d'une variété compacte, il n'y a pas de fonctions harmoniques bornées si et seulement si la vitesse de fuite du mouvement brownien est nulle.

Mariusz Lemanczyk (Torun, Poland) Systems with simple convolutions, distal simplicity and disjointness with infinitely divisible systems We will consider dynamical systems with so called joining primeness property (JPP). These are systems which, whenever joined ergodically with a Cartesian product, are joined with only one "coordinate" system. We show that such systems are disjoint in the sense of Furstenberg with systems which admit "sufficiently many" decompositions into direct products. All systems which are distally simple are JPP, and so are systems whose (reduced) maximal spectral type is singular with respect to the convolution of any two continuous measures. Finally we show that this latter (spectral) property is satisfied for many examples of smooth flows on surfaces.
It is a joint work with F. Parreau and E. Roy.

Ian Melbourne (University of Surrey, UK) Polynomial large deviation estimates for nonuniformly hyperbolic systems We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In particular, if the correlations are summable but decay only at a polynomial rate, then we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal. (In the case of exponential decay of correlations, we obtain the standard exponential large deviation estimates given by a rate function.)
This is joint work with Matthew Nicol.

Florence Merlevède (Paris 6) Déviations modérées pour des suites stationnaires Nous obtenons le principe de déviations modérées pour des suites stationnaires sous des conditions dites projectives. Les résultats s'appliquent en particulier aux fonctions de processus linéaires et aux transformations dilatantes sur l'intervalle.
(Avec J. Dedecker, M. Peligrad et S. Utev.)

Matthew Nicol (University of Houston, USA) Extreme value theory for non-uniformly hyperbolic systems ft: XX is a non-uniformly hyperbolic map (discrete-time) or flow (continuous time ) which may be modelled by a Young tower. Suppose ϕ: XR is a function on X which is locally Holder except for a finite number of singular points. Define Zt(x)=max0≤ sts(x)}. We show that the possible nondegenerate limit distributions for Zt under linear scaling are the type I, II and III distributions of extreme value statistics. We also determine which particular distribution arises (I,II or III) as a function of the regularity of ϕ and the underlying dynamics.

Magda Peligrad (Ohio State University, USA) Limit theorems for weakly associated processes This talk is based on joint works with Sergey Utev and Sunder Sethuraman. The talk will stress the need of a theory of weakly associated processes by pointing out various examples, including the one dimensional nearest-neighbor symmetric exclusion process. Then, it will survey several new techniques involving Rosenthal type maximal inequalities, leading to tightness and weak convergence.

Karl Petersen (Chapel Hill University, USA) Some adic transformations that model nonsimply reinforced random walks Finite graphs lead to stationary adic systems (associated with substitutions and odometers), symbol-count systems (such as the Pascal), and, now also very interesting adic systems that model random walks with general reinforcement schemes. These include the Euler, reverse Euler, and Stirling systems. We explore the connections between these subjects and some dynamical aspects of the adic systems.
(Joint work with Sarah Bailey Frick.)

Emmanuel Rio (Versailles) Vitesses de convergence pour les distances de Wasserstein dans le TLC pour les suites stationnaires Nous obtenons des vitesses de convergence précises dans le TLC pour les suites stationnaires dans L_p pour les distances de Wasserstein d'ordre r, pour p dans ]2,3] et r dans ]p-2,p]. Les conditions proposées sont fondées sur des critères projectifs. Les résultats s'appliquent en particulier à des suites non adaptées.
(avec J. Dedecker et F. Merlévède.)

Emmanuel Roy (Paris 13) Selfjoinings of Poisson suspensions Poisson suspensions offer a canonical way to associate an infinite measure preserving system (the base of the suspension) with a probability preserving system (the suspension). Particularly interesting selfjoinings (called Poisson selfjoinings) are those coming from selfjoinings of the base. We give examples of Poisson suspensions where all ergodic selfjoinings are Poisson. These results allow to precise the structure of factors and to prove disjointness with large classes of dynamical systems. This is a joint work with François Parreau.

Ellen Saada (Rouen) Euler hydrodynamics of one-dimensional attractive systems We consider asymmetric, attractive, irreducible, conservative particle systems on Z with at most K particles per site for which no knowledge of explicit invariant measure is assumed. Typical examples of such processes are the simple exclusion and the misanthrope processes (for which the invariant measures are known) and the K-exclusion processes.
We will focus on the hydrodynamic under Euler scaling of these systems.

Manuel Stadlbauer (University of Göttingen, Germany) On a relative Ruelle-Perron-Frobenius theorem for random countable Markov shifts One of the key steps in the proof of a relative Ruelle-Perron-Frobenius theorem for random countable Markov shifts is to show that a family of conformal measures exists. In this talk, a new construction of an equivariant family of fiber measures will be presented, with emphasis on the contribution of Crauel's relative Prohorov theorem. As an application, we obtain a relative RPF-theorem for random countable topologically mixing Markov shifts (in random environment) with respect to a certain class of locally fiber Hölder continuous functions. This is joint work with Manfred Denker and Yuri Kifer.

Marta Tyran-Kaminska (Katowice, Poland) Poisson limit theorems and convergence to stable laws General methods are known to prove the Central Limit Theorem for the Birkhoff averages of mixing dynamical systems. In particular examples of transformations with a convergence to stable laws has been shown using the method of characteristic function and no general methods are available. The point process theory might offer another method of proof of such convergence. In the last decade it was shown that in many classes of mixing dynamical systems hitting and return times in small sets (balls in metric spaces or cylinder sets), suitably normalized, converge in distribution to exponential law with parameter 1. We are going to combine these techniques to obtain Poisson and stable laws limits in mixing dynamical systems.

Benjamin Weiss (Hebrew University of Jerusalem, Israel) On estimating the entropy from finite samples of noisy data An unknown stationary source produces a signal X1, X2, …, Xn…, which is distorted by random noise. The goal is to give a sequence of estimates (depending on the first n observations) for the entropy of the unknown ergodic process which produced the signal. We will describe some conditions under which this is possible. In the first part of the talk I will survey the entropy estimation methods for uncontaminated data.

Michael Woodroofe (University of Michigan, USA) The Law of the Iterated Logarithm for Stationary Processes The Law of Large Numbers, the Central Limit Theorem, and the Law of the Iterated Logarithm for independent and identically distributed sequences of random variables are three central, perhaps dominant, results of classical probability theory. The Ergodic Theorem provides a complete extension of the Law of Large Numbers to sequences that are dependent, but stationary. The Central Limit Theorem and Law of the Iterated Logarithm do not extend as completely, but only under additional conditions that effectively limit the amount of dependence. During the past decade there has been some progress on understanding the Central Limit Theorem for stationary processes, resulting in conditions that are sufficient and nearly necessary, at least for the conditional version of the Central Limit Theorem. The talk will present recent efforts to modify the arguments leading to the Central Limit Theorem to obtain a Law of the Iterated Logarithm. It will begin with some background material on the Law of the Iterated Logarithm and a selective review of recent work on the Central Limit Theorem for stationary sequences. It will then describe the modifications necessary to obtain the Law of the Iterated Logarithm.

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