Noninvariance of $\bar d$-Convergence of $k$-Step Markov Approximations

Abstract

Among the class of totally ergodic stationary discrete stochastic processes, the Bernoulli processes are characterized by $\bar{d}$-convergence of their canonical $k$-step Markov approximations. Here this property is shown to be no longer invariant under isomorphism if we leave the totally ergodic class. On the other hand, the isomorphism class of all Markov chains is shown to be not closed under $\bar{d}$-convergence.