On Vershikian and I-cosy random variables and filtrations

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.