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

TL;DR

This paper investigates the asymptotic behavior of empirical processes derived from an infinite source Poisson transmission stream, revealing convergence to a non-Gaussian stable process due to long-range dependence.

Contribution

It extends previous work by analyzing non-linear bounded functions and highlights the impact of the transmission rate distribution on asymptotic behavior.

Findings

Normalized fluctuations converge to a non-Gaussian stable process

Long-range dependence influences empirical process behavior

Application to estimating the steady state distribution function

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 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.