Function-indexed empirical processes based on an infinite source Poisson transmission stream (Q442075)

The paper considers an infinite source Poisson transmission process defined by NEWLINE\[NEWLINEX(t) = \sum\limits_{l \in {\mathbb Z}} {{W_l}{{\text{1}}_{[{\Gamma _l} \leqslant t < {\Gamma _l} + {Y_l}]}}},\quad t \in ( - \infty ,\infty ),NEWLINE\]NEWLINEwhere the triples \(\{ ({\Gamma _l},{Y_l},{W_l}),l \in {\mathbb Z}\} \) of session arrival times, durations and transmission rates satisfy the following assumptions:NEWLINE\begin{itemize}NEWLINE\item[(i)] \(\{ {\Gamma _l} : l \in {\mathbb Z}\} \) are the points of a homogeneous Poisson process on the real line;NEWLINE\item[(ii)] \(\{ (Y,W),({Y_l},{W_l}) : l \in {\mathbb Z}\} \) are i.i.d. random pairs with values in \((0,\infty ) \times [0,\infty )\) and independent of the arrival times, the random variable \(W\) is positive with positive probability, and \(Y\) has finite expectation and infinite variance;NEWLINE\item[(iii)] there exists a measure \(\nu \) on \((0,\infty ] \times [0,\infty ]\) such that \(\nu ((0,\infty ] \times [0,\infty ]) = 1\) and, as \(n \to \infty \), \(n \operatorname{P} \left((Y / a(n),W) \in \cdot \right) \xrightarrow{v} \nu \), where \(\xrightarrow{v}\) denotes vague convergence on \((0,\infty ] \times [0,\infty ]\), and \(a\) is the left continuous inverse of \(1 / {\bar F}\) -- here, \(F\) is the distribution function of \(Y\), and \(\bar F = 1 - F\).NEWLINE\end{itemize}NEWLINENEWLINEThe authors study the large time behavior of the empirical process NEWLINE\[NEWLINE{J_T}(\phi ) = \int_0^T {\phi ({X_h}(s))ds} ,NEWLINE\]NEWLINE where \(h > 0\), \({X_h}(s) = \{ X(s + t) : 0 \leqslant t \leqslant h\} \), and \(\phi \) is a real valued function. The main result of the paper is stated as a functional central limit theorem in the Skorohod \({M_1}\) topology.