# On Vershikian and I-Cosy Random Variables and Filtrations

@article{Laurent2011OnVA, title={On Vershikian and I-Cosy Random Variables and Filtrations}, author={St{\'e}phane Laurent}, journal={Theory of Probability and Its Applications}, year={2011}, volume={55}, pages={54-76} }

We prove that the equivalence between Vershik's standardness criterion and the I-cosiness criterion for a filtration in discrete, negative time holds separately for each random variable. This gives a strengthening and a more direct proof of the global equivalence between these two criteria. We also provide more elementary original propositions on Vershik's standardness criterion, while emphasizing that similar statements for I-cosiness are sometimes not so obvious.

## 29 Citations

On Standardness and I-cosiness

- Mathematics
- 2011

The object of study of this work is the invariant characteristics of filtrations in discrete, negative time, pioneered by Vershik. We prove the equivalence between I-cosiness and standardness without…

Vershik’s Intermediate Level Standardness Criterion and the Scale of an Automorphism

- Mathematics
- 2013

In the case of r n -adic filtrations, Vershik’s standardness criterion takes a particular form, hereafter called Vershik’s intermediate level criterion. This criterion, whose nature is combinatorial,…

The filtration of the split-words process

- Mathematics
- 2011

Smorodinsky and Laurent have initiated the study of the filtrations of split-word processes, in the framework of discrete negative time. For these filtrations, we show that Laurent’s sufficient…

Uniform Entropy Scalings of Filtrations

- Computer Science, MathematicsLecture Notes in Mathematics
- 2019

Among the main results, it is proved that the scaled entropy of the filtration generated by the Vershik progressive predictions of a random variable is equal to the scaling entropy of this random variable.

Filtrations at the threshold of standardness

- Mathematics
- 2012

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 of the Erased-Word Processes

- Computer Science
- 2016

It is shown that the poly-adic filtration generated by such a process is standard and constructed with the help of Vershik's theory of filtrations in discrete negative time.

PR ] 1 A ug 2 01 2 Filtrations at the threshold of standardness

- Mathematics
- 2018

A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time −∞, called non-standardness. For example, two dyadic filtrations…

Complementability and Maximality in Different Contexts: Ergodic Theory, Brownian and Poly-Adic Filtrations

- MathematicsLecture Notes in Mathematics
- 2019

The notions of complementability and maximality were introduced in 1974 by Ornstein and Weiss in the context of the automorphisms of a probability space, in 2008 by Brossard and Leuridan in the…

The theory of filtrations of subalgebras, standardness, and independence

- Mathematics
- 2017

This survey is devoted to the combinatorial and metric theory of filtrations: decreasing sequences of -algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of…

Further comments on the representation problem for stationary processes

- Mathematics, Computer Science
- 2010

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