On Martin Boundaries for the Direct Product of Markov Chains

S. A. Molchanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 353-358

Résumé

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 $X^2$.