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Teoriya Veroyatnostei i ee Primeneniya, 1977, Volume 22, Issue 3, Pages 575–581
(Mi tvp3257)
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This article is cited in 16 scientific papers (total in 16 papers)
Short Communications
«Optional times» for random fields
I. V. Evstigneev Moscow
Abstract:
Let $\{\mathscr F_V\}$ be a family of $\sigma$-algebras parametrized by closed subsets $V$ in an $n$-dimensional Euclidean space $X$. Assume that $\{\mathscr F_V\}$ possesses the following properties: (I) if $V'\subseteq V''$, then $\mathscr F_{V'}\subseteq\mathscr F_{V''}$; (II) $\displaystyle\bigcap_{\varepsilon>0}\mathscr F_{V_{\varepsilon}}=\mathscr F_V$, $V_{\varepsilon}$ being the $\varepsilon$-vicinity of $V$. For any random field (usual or generalized), the family of $\sigma$-algebras $\mathscr F_V$ describing the behaviour of the field in the infinitesimal vicinity of $V$ has the above properties.
A random closed set $T(\omega)$ is called optional with respect to the family $\{\mathscr F_V\}$ if $\{\omega\colon T(\omega)\subseteq V\}\in\mathscr F_V$ for all $V$. Such random sets are analogous to optional times in the one-dimensional case. In particular, if the field is Markov, we can prove a version of the strong Markov property with respect to such sets. The result is formulated in terms of $\{\mathscr F_V\}$ only and requires no additional information about the field. Given a usual random field with continous sample functions, a connected component of a level set is an example of a multidimensional «optional time».
Received: 19.11.1975
Citation:
I. V. Evstigneev, “«Optional times» for random fields”, Teor. Veroyatnost. i Primenen., 22:3 (1977), 575–581; Theory Probab. Appl., 22:3 (1978), 563–569
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https://www.mathnet.ru/eng/tvp3257 https://www.mathnet.ru/eng/tvp/v22/i3/p575
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