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Pointwise ergodic theorems with rate and application to the CLT for Markov chains

Christophe Cuny, Michael Lin (2009)

Annales de l'I.H.P. Probabilités et statistiques

Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe....

Position dependent random maps in one and higher dimensions

Wael Bahsoun, Paweł Góra (2005)

Studia Mathematica

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied on each iteration of the process. We study random maps with position dependent probabilities on the interval and on a bounded domain of ℝⁿ. Sufficient conditions for the existence of an absolutely continuous invariant measure for a random map with position dependent probabilities on the interval and on a bounded domain of ℝⁿ are the main results.

Quantitative recurrence in two-dimensional extended processes

Françoise Pène, Benoît Saussol (2009)

Annales de l'I.H.P. Probabilités et statistiques

Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighbourhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including ℤ2-extension of mixing subshifts of finite type. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a result of convergence in...

Quantization Dimension Function and Ergodic Measure with Bounded Distortion

Mrinal Kanti Roychowdhury (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

The quantization dimension function for the image measure of a shift-invariant ergodic measure with bounded distortion on a self-conformal set is determined, and its relationship to the temperature function of the thermodynamic formalism arising in multifractal analysis is established.

Random permutations and unique fully supported ergodicity for the Euler adic transformation

Sarah Bailey Frick, Karl Petersen (2008)

Annales de l'I.H.P. Probabilités et statistiques

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

Sets with doubleton sections, good sets and ergodic theory

A. Kłopotowski, M. G. Nadkarni, H. Sarbadhikari, S. M. Srivastava (2002)

Fundamenta Mathematicae

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

Simple systems are disjoint from Gaussian systems

Andrés del Junco, Mariusz Lemańczyk (1999)

Studia Mathematica

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

Sur la convergence faible des systèmes dynamiques échantillonnés

Nadine Guillotin-Plantard (2004)

Annales de l’institut Fourier

Soit T α la rotation sur le cercle d’angle irrationnel α , soit ( S k ) k ≥ 0 une marche aléatoire transiente sur ℤ . Soit f ∈ L 2 ( μ ) et H ∈ ] 0 , 1 [ , nous étudions la convergence faible de la suite 1 n H ∑ k = 0 [ n t ] - 1 f ∘ T α S k , n ≥ 1 .

Tail fields generated by symbol counts in measure-preserving systems

Karl Petersen, Jean-Paul Thouvenot (2004)

Colloquium Mathematicae

A finite-state stationary process is called (one- or two-sided) super-K if its (one- or two-sided) super-tail field-generated by keeping track of (initial or central) symbol counts as well as of arbitrarily remote names-is trivial. We prove that for every process (α,T) which has a direct Bernoulli factor there is a generating partition β whose one-sided super-tail equals the usual one-sided tail of β. Consequently, every K-process with a direct Bernoulli factor has a one-sided super-K generator....

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

Tim Austin (2016)

Analysis and Geometry in Metric Spaces

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain...

The M/M/1 queue is Bernoulli

Michael Keane, Neil O'Connell (2008)

Colloquium Mathematicae

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. We show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke's theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result...

Topics on Meixner families

Marek Bożejko, Nizar Demni (2010)

Banach Center Publications

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.

Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques

Loïc Hervé (2008)

Annales de l'I.H.P. Probabilités et statistiques

Soit Q une probabilité de transition sur un espace mesurable E, admettant une probabilité invariante, soit (Xn)n une chaîne de Markov associée à Q, et soit ξ une fonction réelle mesurable sur E, et Sn=∑nk=1ξ(Xk). Sous des hypothèses fonctionnelles sur l’action de Q et des noyaux de Fourier Q(t), nous étudions la vitesse de convergence dans le théorème limite central pour la suite ( S n n ) n . Selon les hypothèses nous obtenons une vitesse enn−τ/2 pour tout τ&lt;1, ou bien en n−1/2. Nous appliquons la...

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