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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1963, Volume 8, Issue 4, Pages 451–462 (Mi tvp4692)

Short Communications

Markov Measures and Markov Extensions

N. N. Vorob'ev

Abstract: Let ${\mathfrak{K}}$ be a complex with the set of vertices $M$ and $A$, $B$ and $R$ three subsets of $M$. $R$ is said to be separating $A$ and $B$ in ${\mathfrak{K}}$ (notation: $(A\mathop |\limits_R B)_\mathfrak{K}$) if any $a \in A$ and $b\in B$ are not connected in $\mathfrak{K}\setminus\cup_{r\in R}O_\mathfrak{K}r$ ($O_\mathfrak{K}r$ is the star of $r$ in $\mathfrak{K}$).
Let $S_a,a\in M$, be a finite set and $S_A=\prod_{a\in A}S_a,A\subset M$. A measure $\mu _M$ on $S_M$ is said to be Markov relative to $\mathfrak{K}$ if for any separation $(A\mathop |\limits_R B)_\mathfrak{K}$ and $x_R\in S_R$ the inequality, $\mu _M(x_R)\ne0$ implies
$$\mu _M(X_A\times X_B|x_R) \ne\mu_M(X_A|x_R)\mu_M(X_B|x_R)$$
for arbitrary $X_A\subset S_A$ and $X_B\subset S_B$.
Theorem. If the complex $\mathfrak{K}$ is regular, any consistent family of measures $\mu_\mathfrak{K}=\{ {\mu _K}\}_{K\in\mathfrak{K}}$ on $S_\mathfrak{K}=\{{S_K}\}_{K\in\mathfrak{K}}$ has a unique extension which is Markov relative to $\mathfrak{K}$.

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English version:
Theory of Probability and its Applications, 1963, 8:4, 420–429

Citation: N. N. Vorob'ev, “Markov Measures and Markov Extensions”, Teor. Veroyatnost. i Primenen., 8:4 (1963), 451–462; Theory Probab. Appl., 8:4 (1963), 420–429

Citation in format AMSBIB
\Bibitem{Vor63} \by N.~N.~Vorob'ev \paper Markov Measures and Markov Extensions \jour Teor. Veroyatnost. i Primenen. \yr 1963 \vol 8 \issue 4 \pages 451--462 \mathnet{http://mi.mathnet.ru/tvp4692} \transl \jour Theory Probab. Appl. \yr 1963 \vol 8 \issue 4 \pages 420--429 \crossref{https://doi.org/10.1137/1108047}