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AN IDENTIFICATION PROBLEM IN AN URN AND BALL MODEL WITH HEAVY TAILED DISTRIBUTIONS

Published online by Cambridge University Press:  21 December 2009

Christine Fricker ,
Fabrice Guillemin  and
Philippe Robert
Christine Fricker
Affiliation:
INRIA Paris–Rocquencourt Domaine de Voluceau 78153 Le Chesnay, France E-mail: [email protected]
Fabrice Guillemin
Affiliation:
Orange Labs F-22300 Lannion, France E-mail: [email protected]
Philippe Robert
Affiliation:
NRIA Paris–Rocquencourt Domaine de Voluceau 78153 Le Chesnay, France E-mail: [email protected]
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Abstract

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 77 - 97
Copyright
Copyright © Cambridge University Press 2009

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References

1.Abramowitz, M. & Stegun, I. (1972). Handbook of mathematical functions. National Bureau of Standards Applied Mathematics Series 55. Washington, DC: US Governement Printing Office.Google Scholar
2.Antunes, N., Chabchoub, Y., Fricker, C., Guillemin, F. & Robert, P. (2009). On the statistical characterization of flows in Internet traffic with application to sampling. Computer Communications.Google Scholar
3.Asmussen, S., Klüppelberg, C. & Sigman, K. (1999). Sampling at subexponential times, with queueing application. Stochastic Processes abd Their Application 79: 265–286.CrossRefGoogle Scholar
4.Barbour, A.D., Holst, L. & Janson, S. (1992). Poisson approximation. New York: The Clarendon Press Oxford University Press.CrossRefGoogle Scholar
5.Chabchoub, Y., Fricker, C., Guillemin, F. & Robert, P. (2007). Deterministic versus probabilistic packet sampling in the Internet. In Proceedings of ITC’20.Google Scholar
6.Feller, W. (1966). An introduction to probability theory: Theory and application. Vol. 2, New York: Wiley.Google Scholar
7.Foss, S. & Korshunov, D. (2000). Sampling at a random time with a heavy-tailed distribution. Markov Processes and Related Fields 6(4): 543–568.Google Scholar
8.Robert, P. (2000). Réseaux et files d'attente: Méthodes probabilistes. Mathématiques et Applications. Vol. 35, Berlin: Springer-Verlag.Google Scholar
9.Siegel, A. (2009). Toward a usable theory of chernoff bounds for heterogeneous and partially dependent random variables. Available from http://cs.nyu.edu/faculty/siegel/HHf.pdf.Google Scholar