Leuridan, Christophe

The aim of the present paper is the study of filtrations indexed by the non-positive integers associated to (non-invertible) measure-preserving maps. We establish a necessary and sufficient conditions for the filtration associated to some skew-products to be Kolmogorovian, i.e. to have a trivial tail σ-field at time −∞. This condition inproves on M...

Laurent, Stéphane

We study Vershik and Gorbulsky's notion of entropy scalings for filtrations in the particular case when the scaling is not $\epsilon$-dependent, and is then termed as a uniform scaling. Among our main results, we prove that the scaled entropy of the filtration generated by the Vershik progressive predictions of a random variable is equal to the sca...

Vershik, Anatoly M.
Published in
Japanese Journal of Mathematics

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The main subject is the asymptotic structure of filtrations and a new notion of standardness. All graded graphs and...

Vershik, A. M.
Published in
Functional Analysis and Its Applications

The notion of a homogeneous standard filtration of σ-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables (is Bernoulli), if and only if a standardness criterion is satisfied. The author has recently generalized the notion of...

Laurent, Stéphane

We define a class of erased-word processes and prove that the poly-adic filtration generated by such a process is standard. This is shown by firstly constructing a generating process of innovations in the case of a finite alphabet equipped with the uniform probability measure, and then by deriving the general case with the help of the tools of Vers...

Ceillier, Gaël Leuridan, Christophe

A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time $-\infty$, called non-standardness. For example, two dyadic filtrations with trivial tail $\sigma$-field are not necessarily isomorphic. Yet, any essentially separable filtration indexed by the non-positive integers ...

Laurent, Stéphane

The value of the next-jump time process at each time is the date of its next jump. We characterize the standardness of the filtration generated by this process in terms of the asymptotic behavior at n = −∞ of the probability that the process jumps at time n. In the case when the filtration is not standard we characterize the standardness of its ext...

Araki, Huzihiro
Published in
Letters in Mathematical Physics

For a system of spins and Fermions (satisfying graded commutation relations) on a lattice, a C*-dynamics can be associated with a potential, which has a natural convergence property and a very convenient standardness property. The existence and the uniqueness of the potential with the required properties for any C*-dynamics under consideration, the...