Summary
A second order error bound is obtained for approximating ∫h d \(\tilde Q\) by ∫h d \(\tilde Q\), where\(\tilde Q\) is a convolution of measures andQ a compound Poisson measure on a measurable abelian group, and the functionh is not necessarily bounded. This error bound is more refined than the usual total variation bound in the sense that it contains the functionh. The method used is inspired by Stein's method and hinges on bounding Radon-Nikodym derivatives related to\({{d\tilde Q} \mathord{\left/ {\vphantom {{d\tilde Q} {dQ}}} \right. \kern-\nulldelimiterspace} {dQ}}\). The approximation theorem is then applied to obtain a large deviation result on groups, which in turn is applied to multivariate Poisson approximation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arratia, R., Goldstein, L., Gordon, L.: Poisson approximation and the Chen-Stein method. Statist. Sci.5, 403–434 (1990)
Barbour, A.D.: Stein's method and Poisson process convergence. J. Appl. Probab.25 A, 175–184 (1988)
Barbour, A.D., Chen, L.H.Y., Choi, K.P.: Poisson approximation for unbounded functions, I: independent summands. Statist. Sinica5, 749–766 (1995)
Barbour, A.D., Chen, L.H.Y., Loh, W.L.: Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Probab.20, 1843–1866 (1992)
Barbour, A.D., Hall, P.: On the rate of Poisson convergence. Math. Proc. Camb. Philos. Soc.95, 473–480 (1984)
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Oxford Studies in Probability, vol. 2. Oxford: Clarendon Press 1992
Borovkov, K.A., Pfeifer, D.: Pseudo-Poisson approximation for Markov chains. Preprint (1994)
Chen, L.H.Y.: An approximation theorem for convolutions of probability measures. Ann. Probab.3, 992–999 (1975)
Chen, L.H.Y., Choi, K.P.: Some asymptotic and large deviation results in Poisson approximation. Ann. Probab.20, 1867–1876 (1992)
Le Cam, L.: An approximation theorem for the Poisson binomial distribution. Pacific J. Math.10, 1181–1197 (1960)
Michel, R.: An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. ASTIN Bull.17, 165–169 (1988)
Roos, M.: Stein's method for compound Poisson approximation: the local approach. Ann. Appl. Probab.4, 1177–1187 (1994)
Simons, G., Johnson, N.L.: On the convergence of binomial to Poisson distributions. Ann. Math. Statist.42, 1735–1736 (1971)
Author information
Authors and Affiliations
Additional information
Research of the second author was supported by Schweizerischer Nationalfonds
Rights and permissions
About this article
Cite this article
Chen, L.H.Y., Roos, M. Compound Poisson approximation for unbounded functions on a group, with application to large deviations. Probab. Th. Rel. Fields 103, 515–528 (1995). https://doi.org/10.1007/BF01246337
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01246337