General shot noise processes and functional convergence to stable processes.

*(English)* Zbl 1259.60034
Zili, Mounir (ed.) et al., Stochastic differential equations and processes. SAAP, Tunisia, October 7-9, 2010. Selected papers based on the presentations at the international conference on stochastic analysis and applied probability. Berlin: Springer (ISBN 978-3-642-22367-9/hbk; 978-3-642-22368-6/ebook). Springer Proceedings in Mathematics 7, 151-178 (2012).

The paper investigates the cumulative input process in a system with a unique server dealing with an infinite sized source. The source sends data to the server over independent transmissions and according to a Poisson process. The cumulative input process has a structure of the compound Poisson shot noise that is a natural generalization of the compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. The authors do not assume any particular mechanism of evolution in time of this process (i.e., on the summands). It is shown that the cumulative process, when adequately drifted, rescaled in time and normalized, functionally converges in law to a stable (non-Gaussian) process. This convergence is obtained under two crucial assumptions: (1) the size of each transmission has a regularly varying distribution tail; and (2) an assumption very close to a slow connection rate or slow input rate condition on the length of the transmission. For the entire collection see [

Zbl 1227.60004].

##### MSC:

60F17 | Functional limit theorems; invariance principles |

60G52 | Stable processes |