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Stein’s method and birth-death processes. (English) Zbl 1019.60019
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 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 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 G. V. Weinberg [Stochastic Processes Appl. 87, No. 1, 149-165 (2000)].

MSC:
60F05 Central limit and other weak theorems
60E15 Inequalities; stochastic orderings
60E05 Probability distributions: general theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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