Ratio ergodicity for non-homogeneous Markov chains in general state spaces
Suppose \(T\) is an infinite nonnegative matrix satisfying \(T \text{\textbf{1}} = \text{\textbf{1}}\) (that is, a stochastic matrix) with no zero columns. Then it was shown by the reviewer [ibid. 6, No. 2, 345-352 (1993; Zbl 0774.60013)] that \[ \tau_ 1 (T) \leq \overline \tau_ B (T') \tag \(*\){} \] where \(\tau_ 1\) is Dobrushin's coefficient of ergodicity, and \(\overline \tau_ B\) is Birkhoff's as defined by \textit{S. Gilbert} and \textit{A. Mukherjea} [Math. Z. 196, 485-490 (1987; Zbl 0623.15013)] for a nonnegative matrix \(A\) whose columns are uniformly bounded from zero and infinity, and which has no zero rows, by taking \(A=T'\). The authors extend \((*)\) to a general positive operator \(T\) on a space of measures, where \(T\) preserves the constant function \textbf{1} and has positive adjoint \(T'\), and \(\overline \tau_ B\) is defined, more generally, so that \(\overline \tau_ B(T) = \overline \tau_ B(T')\). A corresponding extension to general state spaces of the application to inhomogeneous Markov chains is made.
scientific article; zbMATH DE number 727273