On Standardness and I-cosiness

Abstract

Summary. 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 using Vershik's standardness criterion. The equivalence between I-cosiness and productness for homogeneous filtrations is further investigated by showing that the I-cosiness criterion is equivalent to Vershik's first level criterion separately for each random variable. We also aim to derive the elementary properties of both these criteria, and to give a survey and some complements on the published and unpublished literature.Acknowledgments. Financial support from the IAP research network (grant nr. P6/03 of the Belgian government, Belgian Science Policy) is gratefully acknowledged. I am also indebted to M. Émery for helpful and encouraging comments and suggestions on earlier drafts of this paper.

On Vershikian and I-cosy random variables and filtrations

“…We use the same notation for L 1 spaces; the space L 1 (C ; E) is the set of all C -measurable random variables X taking their values in E and such that E[ρ(X, x)] is finite for some (⇔ for all) x ∈ E; the space L 1 (C ; E) is endowed with the metric (X, Y ) → E[ρ(X, Y )]. We will implicitly use the fact that m L 1 (B m ; E) is dense in L 1 ( m B m ; E) for any increasing sequence (B m ) m∈N of σ-fields (see Lemma 2.12 in [9]). …”

“…One has Ψ(f (X)) = f (Ψ(X)) for any measurable function f from a Polish space to another one. Details are provided in Annex A of [9]. We will use the following lemmas and propositions from this Annex.…”

“…The three lemmas are copied verbatim from [9]. The third one is a direct consequence of the first two ones.…”

“…A convenient tool for dealing with locally separable filtrations is the existence of a parameterization for such a filtration, stated in Lemma 2.8 which is borrowed from [9]. D e f i n i t i o n 2.2.…”

“…Our motivation to define the general notion of a self-joining criterion is that many elementary properties of I-cosiness are actually true for any self-joining criterion. Two other self-joining criteria are considered in [9], namely Rosenblatt's self-joining criterion and Vershik's self-joining criterion. Contrary to the I-cosiness self-joining criterion, these ones are not hereditary according to Definition 2.6.…”