Convergence of quasi-stationary distributions in birth-death processes
Let \({\mathcal N}(t)\) be an ergodic birth-death process on the positive half- lattice \({\mathcal N}\) and consider the sequence of partitions \({\mathcal N}=G_ k+B_ k\) where \(G_ k=(0,1,...,k)\). Let q(T,k) be the asymptotic distribution of the survivors on \(G_ k\) as \(t\to \infty\), the associated quasi-stationary distribution. Consider the reflecting process \({\mathcal N}^ R_ k(t)\) with reflecting barrier at k and two absorbing processes \({\mathcal N}^ 0_ k(t)\) and \({\mathcal N}^ Q_ k(t)\) where for the former, whenever a sample path reaches the absorbing state \(k+1\), it is replaced at 0 and for the latter it is replaced according to q(T,k). If \(e_ k^{RT}\), \(e_ k^{0T}\), and \(e_ k^{QT}\) are the corresponding ergodic distributions, then the authors show that \(e_ k^{QT}=q(T,k)\) and that \(e_ k^{0T}<e_ k^{QT}<e_ k^{RT}\) for all k where \(''<''\) stands for a partial order ''stochastically less than'' defined in an appendix to the paper. Since both \(e_ k^{0T}\) and \(e_ k^{RT}\) converge to \(e^ t\), the ergodic distribution of the original birth-death process, \(q(T,k)\to e^ T\) as \(t\to \infty\).
scientific article; zbMATH DE number 3890478
scientific article