Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic.
Accepté le :
DOI : 10.1051/ps/2017018
Mots clés : Poisson cluster process, scaling, moments, cumulants, heavy tails, slow growth regime, fast growth regime, Internet traffic modeling
@article{PS_2017__21__369_0,
author = {Antunes, Nelson and Pipiras, Vladas and Abry, Patrice and Veitch, Darryl},
title = {Small and large scale behavior of moments of {Poisson} cluster processes},
journal = {ESAIM: Probability and Statistics},
pages = {369--393},
publisher = {EDP-Sciences},
volume = {21},
year = {2017},
doi = {10.1051/ps/2017018},
mrnumber = {3743919},
zbl = {1393.60051},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2017018/}
}
TY - JOUR AU - Antunes, Nelson AU - Pipiras, Vladas AU - Abry, Patrice AU - Veitch, Darryl TI - Small and large scale behavior of moments of Poisson cluster processes JO - ESAIM: Probability and Statistics PY - 2017 SP - 369 EP - 393 VL - 21 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2017018/ DO - 10.1051/ps/2017018 LA - en ID - PS_2017__21__369_0 ER -
%0 Journal Article %A Antunes, Nelson %A Pipiras, Vladas %A Abry, Patrice %A Veitch, Darryl %T Small and large scale behavior of moments of Poisson cluster processes %J ESAIM: Probability and Statistics %D 2017 %P 369-393 %V 21 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2017018/ %R 10.1051/ps/2017018 %G en %F PS_2017__21__369_0
Antunes, Nelson; Pipiras, Vladas; Abry, Patrice; Veitch, Darryl. Small and large scale behavior of moments of Poisson cluster processes. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 369-393. doi : 10.1051/ps/2017018. http://www.numdam.org/articles/10.1051/ps/2017018/
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