Specifications and a stopping theorem for random fields

S. E. Kuznecov

Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 65-78

Résumé

A family $\mathbf F=\{\mathscr F_t\}_{t\in\mathscr T}$ where $\mathscr T$ is a partially ordered set and $\mathscr F_t$ is a sub-$\sigma$-field of $\mathscr F$, is called a random field if $\mathscr F_s\subseteq\mathscr F_t$ whenever $s\le t$. We consider the problem of existence of compatible conditional distributions (specification) $ p_t(\omega,A)$, $A\in\mathscr F$ for a given random field $\mathbf F$. Let $\mathscr T_0$ be a subset of $\mathscr T$ such that the set $\{t\,:\,t\in\mathscr T_0,\,t>s\}$ is countable for any $s\in\mathscr T$. The set $\mathscr T_0$ is called a skeleton of $\mathbf F$ if for each $t\in\mathscr T_0$ the $\sigma$-field $\mathscr F_t$ is countably generated and for each $t\in\mathscr T\diagdown\mathscr T_0$ one of the following conditions holds: A. There exists a decreasing sequence $t^{(n)}\in\mathscr T_0$, $t^{(n)}\ge t$ such that $\displaystyle\mathscr F_t=\bigcap_n\mathscr F_{t^{(n)}}$. B. There exists an increasing sequence $t_{(n)}\in\mathscr T_0$, $t_{(n)}\le t$ such that: (i) $\displaystyle\mathscr F_t=\bigvee_n\mathscr F_{t_{(n)}}$; (ii) if $s$, $s\in\mathscr T$ than $s$ for some $N$. Theorem 1. Let the $\sigma$-field $\mathscr F$ be countably generated and the measure $\mathbf P$ be perfect. If the random field $\mathbf F$ has a skeleton, than it has a specification. As examples lattice fields, generalized random fields, stochastic processes with $n$-dimensional time etc. are considered. Under some slightly stronger conditions we prove that for any Markov time $\tau(\omega)$ the following stopping theorem holds: $$ \mathbf P(A\mid\mathscr F_\tau)=p_{\tau(\omega)}(\omega,A) \text{ a.\,s. }\mathbf P(A\in\mathscr F). $$ For a Markov field we prove the existence of a Markov specification.