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Estimation of the Memory Parameter of the Infinite Source Poisson Process

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Gilles Fay (LPP), François Roueff (LTCI), Philippe Soulier (MODAL'X)

Statistics Theory · Submitted 16 Sep 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our estimator is shown to be consistent and robust to some form of non-stationarity.

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arXiv IDmath/0509371 (version 3)
Pith IDN5WV5MHZ, Pith's permanent record of this paper and its review
CitationFran\c{c}ois Roueff (LTCI), Gilles Fay (LPP), Philippe Soulier (MODAL'X). "Estimation of the Memory Parameter of the Infinite Source Poisson Process." arXiv:math/0509371 (2005). https://pith.science/paper/N5WV5MHZ
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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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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Long-range dependence induced by heavy tails is a widely reported feature of internet traffic. background 1.00 0.04 0.02 asserted 0.97
s1 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. background 1.00 0.05 0.03 asserted 1.00
s2 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. background 1.00 0.05 0.03 asserted 0.99
s3 The Hurst index of the infinite-source Poisson process is then related to the index of regular variation of the service times. background 0.99 0.11 0.06 asserted 1.00
s4 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. contribution 0.82 0.94 0.24 asserted 0.99
s5 Our estimator is shown to be consistent and robust to some form of non-stationarity. result 0.99 0.93 0.34 asserted 0.99
s6 Its rate of convergence is investigated. result 0.51 0.87 0.06 asserted 0.72

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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 $M/G/\infty$ 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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