Stein's method and birth-death processes
The representation of a distribution \(Q\) on \(\mathbb{Z}_+\) as the equilibrium distribution of a birth and death process allows a particular realization of Stein's method for \(Q\). The probabilistic argument of \textit{A. Xia} [J. Appl. Probab. 36, No. 1, 287-290 (1999; Zbl 0942.60006)] is extended to yield bounds for the resulting `Stein factors': see also those of \textit{G. V. Weinberg} [ibid. 37, No. 4, 1181-1187 (2000; Zbl 0985.60018)]. The approach is also used to refine the very attractive non-uniform bounds for the second differences of the solutions to the Stein equation for a Poisson point process, in the case of \(d_2\)-test functions, which were originally derived by the authors and \textit{G. V. Weinberg} [Stochastic Processes Appl. 87, No. 1, 149-165 (2000)].
scientific article; zbMATH DE number 1906021