Convergence parameter associated with a Markov chain and a family of functions
The definition proposed of a convergence parameter \(R\left( W\right) \) associated with a Markov chain \(X\) with a measurable state space \(\left( E,\mathcal{B}\right) \) and any nonempty set \(W \) of bounded below measurable functions \(f:E\rightarrow \mathbb{R} \), is wider than the well-known definition of a convergence parameter \(R\) in the sense of D. Vere-Jones, R. L. Tweedie or E. Nummelin. Very often, \(R\left( W\right) <\infty \), and there exists a set playing the role of the absorbing set in Nummelin's definition of \(R\). Special attention is paid to the case in which \(E\) is locally compact, \(X\) is a Feller chain on \(E\), and \(W\) coincides with the family \(\varepsilon _{0}^{+}\) of all compactly supported continuous functions \(f\geq 0\left( f\equiv \!\!\!\!\!/0\right) .\) In particular, certain conditions for \(R\left( \varepsilon _{0}^{+}\right) ^{-1} \) to coincide with the norm of an appropriate modification of the chain transition operator are found.
scientific article; zbMATH DE number 5791046
scientific article