## Bernoulli

• Bernoulli
• Volume 13, Number 2 (2007), 473-491.

### Estimation of the memory parameter of the infinite-source Poisson process

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 13, Number 2 (2007), 473-491.

Dates
First available in Project Euclid: 18 May 2007

https://projecteuclid.org/euclid.bj/1179498757

Digital Object Identifier
doi:10.3150/07-BEJ5123

Mathematical Reviews number (MathSciNet)
MR2331260

Zentralblatt MATH identifier
1127.62070

#### Citation

Faÿ, Gilles; Roueff, François; Soulier, Philippe. Estimation of the memory parameter of the infinite-source Poisson process. Bernoulli 13 (2007), no. 2, 473--491. doi:10.3150/07-BEJ5123. https://projecteuclid.org/euclid.bj/1179498757

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