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A simple development of the Thouvenot relative isomorphism theory. (English) Zbl 0551.28023

Let \(X, Y, Z\) be processes with finite state space such that the pair processes \((X,Y)\), \((X,Z)\) are each stationary and ergodic. Thouvenot’s relative isomorphism theorem says that these two pair processes are isomorphic (via stationary codes) if a) \(X\) and \(Y\) are independent; b) \(Y\) is a \(B\) process; c) \((X,Y)\) and \((X,Z)\) have the same entropy rate; and d) the process \(Z\) satisfies a condition called “very weak Bernoulli relative to \(X\)”. The paper under review gives a simple proof of this result. Since the Ornstein isomorphism theorem is a special case of Thouvenot’s result, one has as a consequence a simple proof of Ornstein’s theorem as well.

MSC:

28D20 Entropy and other invariants
28D05 Measure-preserving transformations
60G10 Stationary stochastic processes
94A24 Coding theorems (Shannon theory)
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