On Scaling Limits of Power Law Shot-Noise Fields

Abstract

This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a nondegenerate α-stable limit, as the intensity of the underlying point process goes to infinity. More precisely, finite dimensional distributions are shown to converge, and the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limit the α-stable white noise field. Analogous results are also obtained for the extremal shot-noise field that converges to a Fréchet white noise field. Finally, these results are applied to the modeling and analysis of interference fields in large wireless networks.