Estimation of the Memory Parameter of the Infinite Source Poisson Process

Details Estimation of the Memory Parameter of the Infinite Source Poisson Process Estimation of the Memory Parameter of the Infinite Source Poisson Process

2005-09-16

arxiv.org/abs/math/0509371v3

The Bernoulli Society / IMS Journal "Bernoulli" 13, 2 (2007) 473--491

Mathematics

Statistics Theory

Final version

Scientific paper

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 $M/G/\infty$ 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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