BV coboundaries over irrational rotations
For every irrational rotation we construct a coboundary which is continuous except at a single point where it has a jump, is nondecreasing, and has zero derivative almost everywhere.
A nonergodic version of Gordin's CLT for integrable stationary processes
The central limit problem for strictly stationary sequences [Abstract of thesis]
On the central limit problem for processes of zero entropy
A central limit theorem for non stationary mixing processes
On non-ergodic versions of limit theorems
The author investigates non ergodic versions of several well known limit theorems for strictly stationary processes. In some cases, the assumptions which are given with respect to general invariant measure, guarantee the validity of the theorem with respect to ergodic components of the measure. In other cases, the limit theorem can fail for all ergodic components, while for the original invariant measure it holds.
Constructions of smooth and analytic cocycles over irrational circle rotations
We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk,...
Large deviations for generic stationary processes
Let (Ω,A,μ,T) be a measure preserving dynamical system. The speed of convergence in probability in the ergodic theorem for a generic function on Ω is arbitrarily slow.
Comparison between criteria leading to the weak invariance principle
The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in (1969), the projective criterion introduced by Dedecker in (1998), which was subsequently improved by Dedecker and Rio in (2000) and the condition introduced by Maxwell and Woodroofe in (2000) later improved upon by Peligrad...
Coboundaries in
Uniqueness of a martingale-coboundary decomposition of stationary processes
In the limit theory for strictly stationary processes , the decomposition proved to be very useful; here is a bimeasurable and measure preserving transformation an is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7])....
A cut salad of cocycles
We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.
Contre-exemple dans le théorème central limite fonctionnel pour les champs aléatoires réels
Determinism and martingale decompositions of strictly stationary random processes
Szemerédi theorem implies Furnsternberg theorem
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