Building a Stationary Stochastic Process from a Finite-dimensional Marginal

If A is a finite alphabet, U ⊂ Z, and μU is a probability measure on A that “looks like” the marginal projection of a stationary stochastic process on A D , then can we “extend” μU to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when D = 1, we provide some sufficient conditions and some necessary conditions for μU to be extendible forD > 1, and show that, in general, the problem is not formally decidable. Mathematics Subject Classification Number: Primary: 37A50, 60G10 (Ergodic Theory of Stationary Stochastic Processes) Secondary: 37B10 (Symbolic Dynamics)