Poisson process approximation: From Palm theory to Stein’s method

This exposition explains the basic ideas of Stein’s method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes. 1. Poisson approximation Stein’s method for Poisson approximation was developed by Chen [13] which is based on the following observation: a nonnegative integer valued random variable W follows Poisson distribution with mean λ, denoted as Po(λ), if and only if IE{λf(W + 1)−Wf(W )} = 0 for all bounded f : Z+ → R, where Z+ : = {0, 1, 2, . . .}. Heuristically, if IE{λf(W + 1) − Wf(W )} ≈ 0 for all bounded f : Z+ → R, then L(W ) ≈ Po(λ). On the other hand, as our interest is often on the difference IP(W ∈ A) − Po(λ)(A) = IE[1A(W )− Po(λ)(A)], where A ⊂ Z+ and 1A is the indicator function on A, it is natural to relate the function λf(w + 1)− wf(w) with 1A(w)− Po(λ)(A), leading to the Stein equation: (1) λf(w + 1)− wf(w) = 1A(w)− Po(λ)(A). If the equation permits a bounded solution fA, then IP(W ∈ A)− Po(λ)(A) = IE{λfA(W + 1)−WfA(W )}; and dTV (L(W ),Po(λ)) : = sup A⊂Z+ |IP(W ∈ A)− Po(λ)(A)| = sup A⊂Z+ |IE{λfA(W + 1)−WfA(W )}|. 1Institute for Mathematical Sciences, National University of Singapore, 3 Prince George’s Park, Singapore 118402, Republic of Singapore, e-mail: [email protected] 2Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia, e-mail: [email protected] ∗Supported by research grant R-155-000-051-112 of the National University of Singapore. †Supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems. AMS 2000 subject classifications: primary 60–02; secondary 60F05, 60G55.