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 = 1 Z δ U i , where Z 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], U i ’s are independent and identically distributed random elements on a compact metric space, and U i ’s are independent of Z . 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 ) with n 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 n is often not explicit in the bound.