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Chapitre D'ouvrage Année : 2022

Filtrations associated to some two-to-one transformations

Christophe Leuridan

Résumé

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 Meilijson's result. More specifically, we focus on dyadic filtrations associated to two-to-one maps provided by skew-products, like [Id, T ] or [T −1 , T ]. Determining whether or not these filtrations are product-type (i.e. can be generated by some sequence of independent random variables) is often difficult, although Vershik's criteria provide tools to investigate this question. In this paper, we revisit many classical examples of filtrations associated to two-to-one maps provided by skew-products. The first examples are rather simple and are given as an illustration of Vershik's intermediate criterion. The last two ones are much more involved and yield non-product-type filtrations. Our purpose is to give a more complete and readable presentation of the proofs already existing. MSC Clasification : 37A05,60J05.
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Dates et versions

hal-01987993 , version 1 (21-01-2019)
hal-01987993 , version 2 (06-01-2020)

Identifiants

  • HAL Id : hal-01987993 , version 2

Citer

Christophe Leuridan. Filtrations associated to some two-to-one transformations. Séminaire de Probabilités LI, 2301, Springer, pp.29--89, 2022, Lecture Notes in Mathematics. ⟨hal-01987993v2⟩
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