Compound Poisson process approximation
A. D. Barbour and Marianne Månsson
Source: Ann. Probab.
Volume 30, Number 3
(2002), 1492-1537.
Abstract
Compound Poisson processes are often useful as approximate models, when describing the occurrence of rare events. In this paper, we develop a method for showing how close such approximations are. Our approach is to use Stein's method directly, rather than by way of declumping and a marked Poisson process; this has conceptual advantages, but entails technical difficulties. Several applications are given to illustrate the procedure.
Primary Subjects: 60G55, 62E17
Secondary Subjects: 60F05
Keywords: Extreme values; point processes; compound Poisson process; Stein's method
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1029867135
Digital Object Identifier: doi:10.1214/aop/1029867135
Mathematical Reviews number (MathSciNet):
MR1920275
Zentralblatt MATH identifier:
1033.60059
References
ALDOUS, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
ARRATIA, R., GOLDSTEIN, L. and GORDON, L. (1989). Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Probab. 17 9-25.
BARBOUR, A. D. and BROWN, T. C. (1992). Stein's method and point process approximation. Stochastic Processes Appl. 43 9-31.
BARBOUR, A. D., CHEN, L. H. Y. and LOH, W.-L. (1992). Compound Poisson approximation for non-negative random variables via Stein's method. Ann. Probab. 20 1843-1866.
BARBOUR, A. D. and CHRy SSAPHINOU, O. (2001). Compound Poisson approximation: A user's guide. Ann. Appl. Probab. 11 964-1002.
BARBOUR, A. D., HOLST, L. and JANSON, S. (1992). Poisson Approximation. Oxford Univ. Press.
BARBOUR, A. D., NOVAK, S. Y. and XIA, A. (1999). Compound Poisson approximation for the distribution of extremes. TU Eindhoven, EURANDOM Research Report 99-040.
BARBOUR, A. D. and UTEV, S. (1998). Solving the Stein Equation in compound Poisson approximation. Adv. Appl. Probab. 30 449-475.
BARBOUR, A. D. and UTEV, S. (1999). Compound Poisson approximation in total variation. Stochastic Process. Appl. 82 89-125.
BROWN, T. and XIA, A. (2001). Stein's method and birth-death processes. Ann. Probab. 29 1373-1403.
ERHARDSSON, T. (1999). Compound Poisson approximation for Markov chains using Stein's method. Ann. Probab. 27 565-596.
KALLENBERG, O. (1983). Random Measures, 3rd ed. Akademie, Berlin.
LINDVALL, T. (1992). Lectures on the Coupling Method. Wiley, New York.
MÅNSSON, M. (2000). On compound Poisson approximation for sequence matching. Combin. Probab. Comput. 9 529-548.
NEUHAUSER, C. (1996). A phase transition for the distribution of matching blocks. Combin. Probab. Comput. 5 139-159.
