Stein’s method and Poisson process approximation for a class of Wasserstein metrics
نویسنده
چکیده
Based on Stein’s method, we derive upper bounds for Poisson process approximation in the L1-Wasserstein metric d (p) 2 , which is based on a slightly adapted Lp-Wasserstein metric between point measures. For the case p = 1, this construction yields the metric d2 introduced in [Barbour, A. D. and Brown, T. C. (1992), Stochastic Process. Appl. 43(1), pp. 9–31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general p by showing that d (p) 2 -bounds control differences between expectations of certain p-th order average statistics of point processes.
منابع مشابه
On Stein’s method and perturbations
Stein’s (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often simpler, distribution. In applications of Stein’s method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some St...
متن کاملStein’s method, Palm theory and Poisson process approximation
The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Was...
متن کاملOn Stein’s factors for Poisson approximation in Wasserstein distance
In this note, we provide a probabilistic proof of various Stein’s factors for Poisson approximation in terms of the Wasserstein distance.
متن کاملStein’s Method for Brownian Approximations
Abstract. Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to λ−1/2 where λ is t...
متن کاملPolynomial birth-death distribution approximation in Wasserstein distance
The polynomial birth-death (PBD) distribution on non-negative integers introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates {αi} and death rates {βi}, where αi ≥ 0 and βi ≥ 0 are polynomial functions of i. The family unifies many well-known distributions such as Poisson, negative binomial and binomial. In this talk, I’ll explain how a nice...
متن کامل