Building a Stationary Stochastic Process From a Finite-Dimensional Marginal

Pivato, Marcus

Canadian journal of mathematics, Tome 53 (2001) no. 2, pp. 382-413

Résumé

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.

Détail

DOI : 10.4153/CJM-2001-016-3

Mots-clés : 37A50, 60G10, 37B10

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