On vershikian and I-cosy random variables and filtrations

We prove that the equivalence between Vershik's standardness criterion and the Icosiness 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. © 2011 Society for Industrial and Applied Mathematics.