Strong ergodicity for single-birth processes (Q2731170)

A single-birth process is a continuous-time, irreducible Markov chain on \({\mathbf Z}^{+}\) whose generator \(\{q_{ij}\}\) is such that \(q_{i, i+1} > 0\) and \(q_{i, i+j} = 0\) for all \(i\in {\mathbf Z}^+, j\geq 2\). It is totally stable and conservative if furthermore \(-q_{ii} = \sum_{i\neq j} q_{ij} < +\infty.\) The author gives a necessary, sufficient, and explicit (in terms of \(\{q_{ij}\}\) only) condition for the strong ergodicity (i.e. uniform convergence towards the stationary measure) of a totally stable conservative single-birth process. Applications to the so-called Schögl model are given.