Building a Stationary Stochastic Process From a Finite-dimensional Marginal
Abstract
If A is a finite alphabet, ZD is a D-dimensional lattice, U is a subset of ZD, and muU is a probability measure on AU that ``looks like'' the marginal projection of a stationary random field on A(ZD), then can we ``extend'' muU to such a field? 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 muU to be extendible for D > 1, and show that, in general, the problem is not formally decidable.
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