Poly-adic filtrations, standardness, complementability and maximality

Christophe Leuridan ^[1]
1. [1] Institut Fourier
  
  Institut Fourier
  
  Arrondissement de Grenoble, Francia
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 2, 2017, págs. 1218-1246
Idioma: inglés
DOI: 10.1214/15-AOP1085
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Resumen
- Given some essentially separable filtration (Zn)n≤0(Zn)n≤0 indexed by the nonpositive integers, we define the notion of complementability for the filtrations contained in (Zn)n≤0(Zn)n≤0. We also define and characterize the notion of maximality for the poly-adic sub-filtrations of (Zn)n≤0(Zn)n≤0. We show that any poly-adic sub-filtration of (Zn)n≤0(Zn)n≤0 which can be complemented by a Kolmogorovian filtration is maximal in (Zn)n≤0(Zn)n≤0. We show that the converse is false, and we prove a partial converse, which generalizes Vershik’s lacunary isomorphism theorem for poly-adic filtrations.