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
$$ G=(I+S+S^2+\dots)A\eqno(1) $$
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).

Received: 06.06.1965

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

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