Polynomial birth-death distribution approximation in the Wasserstein distance

Abstract

The polynomial birth-death distribution (abbreviated, PBD) on ℐ={0,1,2,} or ℐ={0,1,2,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373-1403, 2001) is the equilibrium distribution of the birth-death process with birth rates {α i } and death rates {β i }, where α i ≥0 and β i ≥0 are polynomial functions of i ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with α i =a and β i =i+bi(i-1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373-1403, 2001) and generalizes the work of ..