Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T21:14:07.079Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rate of Poisson process approximation to a Bernoulli process

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Aihua Xia
Aihua Xia*
Affiliation:
University of New South Wales
*
∗Postal address: Department of Statistics, School of Mathematics, The University of New South Wales, Sydney 2052, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

Keywords

IMMIGRATION–DEATH PROCESSSTEIN'S METHODPALM DISTRIBUTIONSCOUPLING

MSC classification

Primary: 60G55: Point processes 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 898 - 907
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported by a 1995 Australia Research Council Small Grant from the University of New South Wales.

References

Arratia, R., Goldstein, L. and Gordon, L. (1989) Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Barbour, A. D. (1988) Stein's method and Poisson process convergence. J. Appl. Prob. 25, 175–184.CrossRefGoogle Scholar
Barbour, A. D. and Brown, T. C. (1992) Stein's method and point process approximation. Stoch. Proc. Appl. 43, 9–31.CrossRefGoogle Scholar
Barbour, A. D., Brown, T. C. and Xia, A. (1997) Point Processes in Time and Stein's Method. To appear.Google Scholar
Barbour, A. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Cambridge Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press, Oxford.CrossRefGoogle Scholar
Brown, T. C. and Xia, A. (1995a) On metrics in point process approximation. Stoch. Stoch. Rep. 52, 247–263.Google Scholar
Brown, T. C. and Xia, A. (1995b) On Stein-Chen factors for Poisson approximation. Statist. Prob. Lett. 23, 327–332.CrossRefGoogle Scholar
Chen, L. H. Y. (1975) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer, Berlin.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, New York.Google Scholar
Kallenberg, O. (1983) Random Measures. Academic Press, New York.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and their Statistical Inference. Dekker, New York.Google Scholar
Rachev, S. T. (1984) The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Prob. Appl. 29, 647–676.CrossRefGoogle Scholar