**Walter Schachermayer**

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## On Vershik's Standardness Criterion and Tsirelson's Notion of Cosiness.

**M. Emery, W. Schachermayer**

Séminaire de Probabilités XXXV, Springer Lecture Notes in Mathematics, Vol. 1755 (2001), pp. 265-305.

### Abstract:

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. Feldman, M.
Smorodinsky, B. Tsirelson. Key concepts are the notions of standard
filtrations (due to A. Vershik) and cosy filtrations (due to B.
Tsirelson). We investigate the relation between these two concepts and
try to provide a comprehensive and selfcontained presentation of the
topic.

Part of this work is expository, and consists in translating into a
probabilist's language Vershik's necessary and sufficient condition for
standardness, and his theorem on lacunary isomorphism. There are also
original results: Theorem 2 proves that standardness is in fact equivalent
to a certain variant of the notion of cosiness, which we call Icosiness;
an example borrowed from Vershik and Smorodinsky then shows that
Icosiness is strictly stronger than another variant, Dcosiness, used in
earlier works. Another new result is a (negative) answer to a question of
H. von Weizsaecker: the last section gives an example of a filtration
(F_n)_{n \in -N} and a \sigmafield B such that F_0 and B are 'almost
independent', but nevertheless

### Preprints:

[PostScript (2355 k)] [PS.gz (1140 k)] [PDF (401 k)] [DOI: 10.1007/978-3-540-44671-2_20]

Publications marked with have appeared in refereed journals.

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