A direct proof that vwb processes are closed in the $$\bar d$$ -metric

Abstract

As defined in the literature, a process is very weak Bernoulli if a certain propertyP(e) is satisfied for everye>0. By means of an easy proof, it is shown that givene>0, there existsδ>0 such that given any two stationary processes whose\(\bar d\)-distance is less thanδ, if one of the processes is very weak Bernoulli then the other process is “almost” very weak Bernoulli in the sense that the propertyP(e) is satisfied. Using this result a direct proof can be given that the very weak Bernoulli processes are closed under the\(\bar d\)-distance, and also that a finitely determined process is very weak Bernoulli. Relativized versions of these results are also considered.