Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-14T06:04:00.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Stein's Method for Compound Geometric Approximation

Part of: Markov processes Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

Fraser Daly
Fraser Daly*
Affiliation:
Universität Zürich
*
∗Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: fraser.daly@math.uzh.ch
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

Keywords

Stein's methodcompound geometric approximationMarkov chain hitting times

MSC classification

Secondary: 62E17: Approximations to distributions (nonasymptotic) 60E15: Inequalities; stochastic orderings 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 146 - 156
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Barbour, A. D. and Chen, L. H. Y. (eds) (2005). An Introduction to Stein's Method. World Scientific, Singapore.Google Scholar
[2] Barbour, A. D. and Grübel, R. (1995). The first divisible sum. J. Theoret. Prob. 8, 39–47.Google Scholar
[3] Barbour, A. D. and Utev, S. (1998). {Solving the Stein equation in compound Poisson approximation}. Adv. Appl. Prob. 30, 449–475.CrossRefGoogle Scholar
[4] Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992). Compound Poisson approximation for nonnegative random variables using Stein's method. Ann. Prob. 20, 1843–1866.Google Scholar
[5] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
[6] Erhardsson, T. (1999). {Compound Poisson approximation for Markov chains using Stein's method}. Ann. Prob. 27, 565–596.Google Scholar
[7] Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Vol. 1. John Wiley, New York.Google Scholar
[8] Griffeath, D. (1975). A maximal coupling for Markov chains. Z. Wahrscheinlichkeitsth 31, 95–106.Google Scholar
[9] Jury, E. I. (1964). Theory and Application of the z-Transform Method. John Wiley, New York.Google Scholar
[10] Just, E. and Schaumberger, N. (1964). Contour integration for rational functions. Amer. Math. Monthly 71, 546–547.Google Scholar
[11] Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht.Google Scholar
[12] Lang, S. (1999). Complex Analysis, 4th edn. Springer, New York.Google Scholar
[13] Peköz, E. A. (1996). {Stein's method for geometric approximation}. J. Appl. Prob. 33, 707–713.CrossRefGoogle Scholar
[14] Phillips, M. J. and Weinberg, G. V. (2000). Non-uniform bounds for geometric approximation. Statist. Prob. Lett. 49, 305–311.Google Scholar
[15] Pitman, J. W. (1976). {On coupling of Markov chains}. Z. Wahrscheinlichkeitsth 35, 315–322.Google Scholar
[16] Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
[17] Stein, C. (1986). Approximate Computation of Expectations (IMS Lecture Notes Monogr. Ser. 7). Institute of Mathematical Statistics, Hayward, CA.Google Scholar