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Building a Stationary Stochastic Process From a Finite-dimensional Marginal

Marcus Pivato
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Abstract

If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of Z^D, and mu_U is a probability measure on A^U that ``looks like'' the marginal projection of a stationary random field on A^(Z^D), then can we ``extend'' mu_U 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 mu_U to be extendible for D > 1, and show that, in general, the problem is not formally decidable.

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Marcus Pivato. 2001-08-12. Building a Stationary Stochastic Process From a Finite-dimensional Marginal. https://arxiv.org/abs/math/0108081

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