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}
}
  • S. Laurent
  • Published 7 March 2011
  • Mathematics
  • Theory of Probability and Its Applications
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. 
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References

SHOWING 1-10 OF 45 REFERENCES
On Standardness and I-cosiness
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…
On Vershik’s Standardness Criterion and Tsirelson’s Notion of Cosiness
Building on work done by A. Vershik some thirty years ago, the insight into different types of filtrations has recently seen important progress, due in particular to B. Tsirelson, and L. Dubins, J.…
Foundations of modern probability (2nd edn), by Olav Kallenberg. Pp. 638. £49 (hbk). 2002. ISBN 0 387 95313 2 (Springer-Verlag).
distribution, queuing theory, random walks, and so on. On many topical issues he is prepared to admit that there are no definitive answers, considering, inter alia, the following questions: how…
Measure Theory
Convergence of Probability Measures
Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
Espaces probabilisés filtrés : de la théorie de Vershik au mouvement brownien, via des idées de Tsirelson
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and…
Foundations of Modern Probability
  • O. Kallenberg
  • Mathematics
    Probability Theory and Stochastic Modelling
  • 2021
* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit…
Triple Points: From Non-Brownian Filtrations to Harmonic Measures
Abstract. ((Without abstract))
Springer-Verlag, 2001, p
  • 265–305
  • 1755
Теория убывающих последовательностей измеримых разбиений
  • Алгебра и анализ,
  • 1994
...
...