The Annals of Probability

Poly-adic filtrations, standardness, complementability and maximality

Christophe Leuridan

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Given some essentially separable filtration $(\mathcal{Z}_{n})_{n\le0}$ indexed by the nonpositive integers, we define the notion of complementability for the filtrations contained in $(\mathcal{Z}_{n})_{n\le0}$. We also define and characterize the notion of maximality for the poly-adic sub-filtrations of $(\mathcal{Z}_{n})_{n\le0}$. We show that any poly-adic sub-filtration of $(\mathcal{Z}_{n})_{n\le0}$ which can be complemented by a Kolmogorovian filtration is maximal in $(\mathcal{Z}_{n})_{n\le0}$. We show that the converse is false, and we prove a partial converse, which generalizes Vershik’s lacunary isomorphism theorem for poly-adic filtrations.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 1218-1246.

Received: July 2014
Revised: December 2015
First available in Project Euclid: 31 March 2017

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces

Filtrations indexed by the nonpositive integers product-type filtrations standard filtrations complementability maximality exchange property


Leuridan, Christophe. Poly-adic filtrations, standardness, complementability and maximality. Ann. Probab. 45 (2017), no. 2, 1218--1246. doi:10.1214/15-AOP1085.

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