A polynomial birth–death point process approximation to the Bernoulli process

We propose a class of polynomial birth–death point processes   (abbreviated to PBDP) Z≔∑i=1ZδUi, where ZZ is a polynomial birth–death random variable defined in [T.C. Brown, A. Xia, Stein’s method and birth–death processes, Ann. Probab. 29 (2001) 1373–1403], UiUi’s are independent and identically distributed random elements on a compact metric space, and UiUi’s are independent of ZZ. We show that, with two appropriately chosen parameters, the error of PBDP approximation to a Bernoulli process is of the order O(n−1/2)O(n−1/2) with nn being the number of trials in the Bernoulli process. Our result improves the performance of Poisson process approximation, where the accuracy is mainly determined by the rarity (i.e. the success probability) of the Bernoulli trials and the dependence on sample size nn is often not explicit in the bound.