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Epsilon-independence between two processes

Tomasz Downarowicz, Paulina Grzegorek (2008)

Studia Mathematica

We study the notion of ε-independence of a process on finitely (or countably) many states and that of ε-independence between two processes defined on the same measure preserving transformation. For that we use the language of entropy. First we demonstrate that if a process is ε-independent then its ε-independence from another process can be verified using a simplified condition. The main direction of our study is to find natural examples of ε-independence. In case of ε-independence of one process,...

Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings

Y. Derriennic, K. Frączek, M. Lemańczyk, F. Parreau (2008)

Colloquium Mathematicae

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...

Ergodicity and conservativity of products of infinite transformations and their inverses

Julien Clancy, Rina Friedberg, Indraneel Kasmalkar, Isaac Loh, Tudor Pădurariu, Cesar E. Silva, Sahana Vasudevan (2016)

Colloquium Mathematicae

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product T × T - 1 is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

Error rates in the Darling-Kac law

Dalia Terhesiu (2014)

Studia Mathematica

This work provides rates of convergence in the Darling-Kac law for infinite measure preserving Pomeau-Manneville (unit interval) maps. Along the way we obtain error rates for the stable law associated with the first return map and the first return time to some suitable set inside the unit interval.

Exponential inequalities and functional central limit theorems for random fields

Jérôme Dedecker (2001)

ESAIM: Probability and Statistics

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform φ -mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients....

Exponential inequalities and functional central limit theorems for random fields

Jérôme Dedecker (2010)

ESAIM: Probability and Statistics

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients. ...

Finitarily Bernoulli factors are dense

Stephen Shea (2013)

Fundamenta Mathematicae

It is not known if every finitary factor of a Bernoulli scheme is finitarily isomorphic to a Bernoulli scheme (is finitarily Bernoulli). In this paper, for any Bernoulli scheme X, we define a metric on the finitary factor maps from X. We show that for any finitary map f: X → Y, there exists a sequence of finitary maps fₙ: X → Y(n) that converges to f, where each Y(n) is finitarily Bernoulli. Thus, the maps to finitarily Bernoulli factors are dense. Let (X(n)) be a sequence of Bernoulli schemes such...

Gaussian automorphisms whose ergodic self-joinings are Gaussian

Mariusz Lemańczyk, F. Parreau, J. Thouvenot (2000)

Fundamenta Mathematicae

 We study ergodic properties of the class of Gaussian automorphisms whose ergodic self-joinings remain Gaussian. For such automorphisms we describe the structure of their factors and of their centralizer. We show that Gaussian automorphisms with simple spectrum belong to this class.  We prove a new sufficient condition for non-disjointness of automorphisms giving rise to a better understanding of Furstenberg's problem relating disjointness to the lack of common factors. This...

Identification of periodic and cyclic fractional stable motions

Vladas Pipiras, Murad S. Taqqu (2008)

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

We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...

Infinite products of random matrices and repeated interaction dynamics

Laurent Bruneau, Alain Joye, Marco Merkli (2010)

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

Let Ψn be a product of n independent, identically distributed random matrices M, with the properties that Ψn is bounded in n, and that M has a deterministic (constant) invariant vector. Assume that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish. We show that Ψn is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as n→∞. The fluctuating part converges in Cesaro mean to a limit that is characterized...

Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique

Jean-Pierre Conze, Yves Guivarc'h (2000)

Colloquium Mathematicae

The invariant measures for a Markovian operator corresponding to a random walk, in a random stationary one-dimensional environment defined by a dynamical system, are quasi-invariant measures for the system. We discuss the construction of such measures in the general case and show unicity, under some assumptions, for a rotation on the circle.

Masse des pointes, temps de retour et enroulements en courbure négative

Nathanaël Enriquez, Jacques Franchi (2002)

Bulletin de la Société Mathématique de France

Soient Γ un groupe discret géométriquement fini d’isométries d’une variété de Hadamard pincée X et 𝒫 une pointe de l’orbifold associé ℳ : = Γ ∖ X . Munissant T 1 ℳ de sa mesure de Patterson-Sullivan m , nous obtenons une estimation asymptotique de la masse d’un petit voisinage horocyclique de 𝒫 , moyennant une hypothèse sur la croissance du sous-groupe parabolique associé à 𝒫 , hypothèse qui est réalisée si X est symétrique de rang 1 . Nous en déduisons une estimation asymptotique du temps de retour du flot géodésique...

Numerical solution of a stochastic model of a ball-type vibration absorber

Fischer, Cyril, Náprstek, Jiří (2021)

Programs and Algorithms of Numerical Mathematics

The mathematical model of a ball-type vibration absorber represents a non-linear differential system which includes non-holonomic constraints. When a random ambient excitation is taken into account, the system has to be treated as a stochastic deferential equation. Depending on the level of simplification, an analytical solution is not practicable and numerical solution procedures have to be applied. The contribution presents a simple stochastic analysis of a particular resonance effect which can...

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