Small and large scale behavior of the Poissonized telecom process
Let \(Y\left( t\right) =\int_X\int_R\int_R\left( G\left( x,t+u\right) -G\left( x,u\right) \right) w\tilde{N}\left( dx,du,dw\right) \) be the Poissonized mixed moving average, where \(\tilde{N}\) is a compensated Poisson point measure. Conditions are obtained for convergence in distribution of \(\lim_{\rho \rightarrow \infty }\rho ^{-H_{\infty }\left( t\right) }Y\left( \rho t\right) \) to the fractional Brownian motion \(B_{H_{\infty }}\left( t\right).\) In case \(\rho \downarrow 0\), a similar convergence problem is considered for \(\rho ^{-H_{\infty }}\left( Y\left( t_{0}+\rho t\right) -Y\left( t_{0}\right) \right) \) and the limit is the stable mixed moving average. The results are applied both to the standard and the canonical Poissonized telecom processes. Some comments on the difference in the similarity parameter \(H_{\infty }\) of these processes are presented.
scientific article; zbMATH DE number 2133630