Polynomial birth-death distribution approximation in the Wasserstein distance (Q1028626)

The polynomial birth-death distribution (PBD) on \(\mathcal{I}=\{ 0,1,2,\dots\}\) or \(\mathcal{I}=\{0,1,2,\dots,m\}\) for some finite \(m\) has been introduced by \textit{T. C. Brown} and \textit{A. Xia} [Ann. Probab. 29, No. 3, 1373--1403 (2001; Zbl 1019.60019)] as the equilibrium distribution of the birth-death process with birth rates \(\{\alpha _{i}\}\) and death rates \(\{\beta _{i}\}\), where \(\alpha_{i}\geq 0\) and \(\beta _{i}\geq 0\) are polynomial functions of \(i\in \mathcal{I}\). It includes the Poisson, negative binomial, binomial, and hypergeometric distributions. The present authors give probabilistic proofs of various Stein's factors for the PBD approximation with \(\alpha _{i}=a\) and \(\beta _{i}=i+bi( i-1)\) in terms of the Wasserstein distance. The results complement the work of Brown and Xia (loc. cit.) and generalize the work of \textit{A. D. Barbour} and \textit{A. Xia} [Bernoulli 12, No. 6, 943--954 (2006; Zbl 1328.62076)], where the Poisson approximation \((b=0)\) in the Wasserstein distance is investigated. As an application, the authors establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.