Function-indexed empirical processes based on an infinite source Poisson transmission stream
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 non-linear 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.
grant W911NF-07-1-0078 at Cornell University, Department of
Mathematics, Universite de Paris Ouest Nanterre during his visit in
2007, Laboratory of
Actuarial Mathematics, Department of Mathematics, University of
Copenhagen and by Laboratory of Informatics, Technical University of
Denmark, during his sabbatical stay in 2008-2009