Bernoulli

  • Bernoulli
  • Volume 18, Number 3 (2012), 783-802.

Function-indexed empirical processes based on an infinite source Poisson transmission stream

François Roueff, Gennady Samorodnitsky, and Philippe Soulier

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Abstract

We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on nonlinear bounded functions exhibit the influence of the limit transmission rate distribution at high session lengths on the asymptotic behavior of the empirical process. As an illustration, we apply the main result to estimation of the distribution function of the steady state value of the transmission process.

Article information

Source
Bernoulli, Volume 18, Number 3 (2012), 783-802.

Dates
First available in Project Euclid: 28 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1340887002

Digital Object Identifier
doi:10.3150/11-BEJ367

Mathematical Reviews number (MathSciNet)
MR2948901

Zentralblatt MATH identifier
1259.60036

Keywords
empirical process long range dependence M/G queue shot noise

Citation

Roueff, François; Samorodnitsky, Gennady; Soulier, Philippe. Function-indexed empirical processes based on an infinite source Poisson transmission stream. Bernoulli 18 (2012), no. 3, 783--802. doi:10.3150/11-BEJ367. https://projecteuclid.org/euclid.bj/1340887002


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