Building a Stationary Stochastic Process From a Finite-dimensional Marginal

2001/08/12 by Marcus Pivato
Mathematics · #math.PR #math.DS #msc:37A50 #msc:60G10 #msc:37B10

paper · pdf

published as Canadian Journal of Mathematics, Vol. 53 (2), 2001 pp.382-413 · LaTeX2E format, 40 pages, 4 figures (eps format)

arxiv created 2001/08/12 · arxiv updated 2009/11/30

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<sup>D</sup>), 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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