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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

Leningrad

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

Received: 08.01.1962

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}


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