Open Access
May 2007 Estimation of the memory parameter of the infinite-source Poisson process
Gilles Faÿ, François Roueff, Philippe Soulier
Bernoulli 13(2): 473-491 (May 2007). DOI: 10.3150/07-BEJ5123


Long-range dependence induced by heavy tails is a widely reported feature of internet traffic. Long-range dependence can be defined as the regular variation of the variance of the integrated process, and half the index of regular variation is then referred to as the Hurst index. The infinite-source Poisson process (a particular case of which is the $M/G/∞$ queue) is a simple and popular model with this property, when the tail of the service time distribution is regularly varying. The Hurst index of the infinite-source Poisson process is then related to the index of regular variation of the service times. In this paper, we present a wavelet-based estimator of the Hurst index of this process, when it is observed either continuously or discretely over an increasing time interval. Our estimator is shown to be consistent and robust to some form of non-stationarity. Its rate of convergence is investigated.


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Gilles Faÿ. François Roueff. Philippe Soulier. "Estimation of the memory parameter of the infinite-source Poisson process." Bernoulli 13 (2) 473 - 491, May 2007.


Published: May 2007
First available in Project Euclid: 18 May 2007

zbMATH: 1127.62070
MathSciNet: MR2331260
Digital Object Identifier: 10.3150/07-BEJ5123

Keywords: heavy tails , Internet traffic , long-range dependence , Poisson point processes , Semiparametric estimation , Wavelets

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability

Vol.13 • No. 2 • May 2007
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