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Building a Stationary Stochastic Process From a Finite-Dimensional Marginal

Published online by Cambridge University Press:  20 November 2018

Marcus Pivato
Marcus Pivato*
Affiliation:
Department of Mathematics University of Toronto 100 St. George Street Toronto, Ontario M5S 3G3, e-mail: [email protected]
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Abstract

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If $\mathfrak{A}$ is a finite alphabet, $\mathcal{U}\,\subset \,{{\mathbb{Z}}^{D}}$, and ${{\mu }_{\mathcal{U}}}$ is a probability measure on ${{\mathfrak{A}}^{\mathcal{U}}}$ that “looks like” the marginal projection of a stationary stochastic process on ${{\mathfrak{A}}^{{{\mathbb{Z}}^{D}}}}$, then can we “extend” ${{\mu }_{\mathcal{U}}}$ 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 ${{\mu }_{\mathcal{U}}}$ to be extendible for $D\,>\,1$, and show that, in general, the problem is not formally decidable.

Keywords

37A5060G1037B10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 2 , 01 April 2001 , pp. 382 - 413
Copyright
Copyright © Canadian Mathematical Society 2001

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