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Building a Stationary Stochastic Process From a Finite-Dimensional Marginal

  Published:2001-04-01
 Printed: Apr 2001
  • Marcus Pivato
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Abstract

If $\mathfrak{A}$ is a finite alphabet, $\sU \subset\mathbb{Z}^D$, and $\mu_\sU$ is a probability measure on $\mathfrak{A}^\sU$ that ``looks like'' the marginal projection of a stationary stochastic process on $\mathfrak{A}^{\mathbb{Z}^D}$, then can we ``extend'' $\mu_\sU$ 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 $\mu_\sU$ to be extendible for $D>1$, and show that, in general, the problem is not formally decidable.
MSC Classifications: 37A50, 60G10, 37B10 show english descriptions Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
Stationary processes
Symbolic dynamics [See also 37Cxx, 37Dxx]
37A50 - Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
60G10 - Stationary processes
37B10 - Symbolic dynamics [See also 37Cxx, 37Dxx]
 

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