Strongly monotone \(q\)-functions and a note on strong ergodicity of monotone \(q\)-functions
A transition function \( P\left( t\right) =(p_{ij}\left( t\right) )_{i,j\in \mathbb{N}},t\geq 0\), is said to be strongly monotone if \(\sum_{k\geq j}p_{ik}\left( t\right) \) and \( p_{ij}\left( t\right) \rightarrow 0\) as \(i\rightarrow \infty \) for any \(j\in \mathbb{N}\) and \(t\geq 0\). For a stable \(q\)-matrix \(Q\), the author gives a necessary and sufficient condition for the minimal \(Q\)-function \(P\left( t\right) \) to be strongly monotone. This yields a criterion for a Feller minimal monotone chain to be strongly ergodic. Finally, an application to birth-and-death processes with killing is given.
scientific article; zbMATH DE number 5144924