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

 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 472–482 (Mi tvp641)

Principles of potential theory and Markov chains

D. I. Shparo

Moscow

Abstract: We consider an infinite matrix $G$ with nonnegative entries and with all its columns tending to 0. We investigate those properties of $G$ which allow us to express $G$ in the form
$$G=(I+S+S^2+…)A\eqno(1)$$
where $I$ is the identity matrix, $S$ is a substochastic matrix and $A$ is a diagonal matrix with positive entries on the diagonal. These properties are
1) $G$ is nondegenerate in a sense,
2) the vector $e$ with all its components equal to 1 is the limit of an increasing sequence of the potentials of nonnegative measures,
3) the principle of domination holds. These properties are also necessary for representation (1).

Full text: PDF file (766 kB)

English version:
Theory of Probability and its Applications, 1966, 11:3, 415–424

Bibliographic databases:

Citation: D. I. Shparo, “Principles of potential theory and Markov chains”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 472–482; Theory Probab. Appl., 11:3 (1966), 415–424

Citation in format AMSBIB
\Bibitem{Shp66} \by D.~I.~Shparo \paper Principles of potential theory and Markov chains \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 3 \pages 472--482 \mathnet{http://mi.mathnet.ru/tvp641} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=203816} \zmath{https://zbmath.org/?q=an:0202.47804} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 3 \pages 415--424 \crossref{https://doi.org/10.1137/1111040}