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We construct a large family of ergodic non-Markovian processes with infinite memory having the same p-dimensional marginal laws of an arbitrary ergodic Markov chain or projection of Markov chains. Some of their spectral and mixing properties are given. We show that the Chapman–Kolmogorov equation for the ergodic transition matrix is generically satisfied by infinite memory processes.

A family of stationary processes with infinite memory having the same p-marginals. Ergodic and spectral properties