Stein's method and Poisson process approximation for a class of Wasserstein metrics

Details Stein's method and Poisson process approximation for a class of Wasserstein metrics Stein's method and Poisson process approximation for a class of Wasserstein metrics

2007-06-08

arxiv.org/abs/0706.1172v3

Bernoulli 2009, Vol. 15, No. 2, 550-568

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/08-BEJ161 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ161

Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 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 $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.

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