Stein's method and poisson process convergence

Abstract

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

On Steinʼs method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

“…For further details, we refer the reader to [1,12,17]. Secondly, combining [27,Theorem 2.7] with the idea of Kuelbs in [15], we will present a relationship among the notions of Gaussian measure on a Banach space, reproducing kernel Hilbert space, and abstract Wiener space.…”

“…When B is the Euclidean space R n , A Z is known as a Stein operator for the standard (multivariate) normal distribution. Among many of approaches to find a Stein operator for one-dimensional distributions, the generator approach developed by Barbour [1,2] seems to be readily extended to our infinite-variate distribution μ Z . Based on Barbour's idea, we proceed with finding the characterizing operator A Z as follows.…”

“…So far, the scope of his discovery has expanded rapidly by Chen [6], Barbour [1,2], and many other authors, which showed that Stein's method can be adapted to approximation by a broad class of probability distributions such as the Poisson, compound Poisson, and Gamma E-mail address: hhshih@nuk.edu.tw. 1 Research supported by the National Science Council of Taiwan. distributions, as well as extensively applied in a wide range of other fields such as the theory of random graphs, computational molecular biology, etc.…”

“…distributions, as well as extensively applied in a wide range of other fields such as the theory of random graphs, computational molecular biology, etc. For further details, see [1][2][3]6,29,30,32] and references cited therein.…”

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

“…On the other hand, the lower bound in (1) tells us that, unlike the normal approximation where the accuracy of approximation improves when the sample size increases, the accuracy of Poisson approximation does not become better when the sample size gets larger. As generalizations of Poisson approximation, the precision of Poisson and compound Poisson process approximations is also basically determined by the rarity of the events in the process being approximated and does not improve as the information accumulates (see [1,2,5,25,14]). …”