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Teoriya Veroyatnostei i ee Primeneniya, 1967, Volume 12, Issue 2, Pages 353–358 (Mi tvp713)  

This article is cited in 11 scientific papers (total in 11 papers)

Short Communications

On Martin Boundaries for the Direct Product of Markov Chains

S. A. Molchanov

Moscow
Abstract: Let $X^i$ be denumerable Markov chains in state spaces $E^i$ with transition matrices $P^i$ $(i=1,2)$. A function $f(x^1,x^2)$ ($x^1\in E^1$, $x^2\in E^2$) is harmonic for chain $X^1\times X^2$ if
$$ (P^1\times P^2)f=f. $$
It is proved that every minimal harmonic function for chain $X^1\times X^2$ may be represented in the form
$$ f(x^1,x^2)=\varphi(x^1)\psi(x^2) $$
where functions $\varphi(x^1)$ and $\psi(x^2)$ are such that
$$ \begin{matrix} P^1\varphi&=&\alpha\varphi&& \\ &&\alpha\beta&=&1 \\ P^2\psi&=&\beta\psi&& \end{matrix} $$
In this way the Martin boundary for chain $X^1\times X^2$ is described in terms of the Martin boundaries for chains $X^1$ and $X^2$.
Received: 01.03.1966
English version:
Theory of Probability and its Applications, 1967, Volume 12, Issue 2, Pages 307–310
DOI: https://doi.org/10.1137/1112035
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: S. A. Molchanov, “On Martin Boundaries for the Direct Product of Markov Chains”, Teor. Veroyatnost. i Primenen., 12:2 (1967), 353–358; Theory Probab. Appl., 12:2 (1967), 307–310
Citation in format AMSBIB
\Bibitem{Mol67}
\by S.~A.~Molchanov
\paper On Martin Boundaries for the Direct Product of Markov Chains
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 2
\pages 353--358
\mathnet{http://mi.mathnet.ru/tvp713}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=216587}
\zmath{https://zbmath.org/?q=an:0308.60044|0292.60119}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 2
\pages 307--310
\crossref{https://doi.org/10.1137/1112035}
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  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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