On Scaling Limits of Power Law Shot-noise Fields

TL;DR

This paper investigates the scaling limits of power law shot-noise fields on Poisson point processes, demonstrating convergence to stable and Fréchet white noise fields, with applications to wireless network analysis.

Contribution

It establishes the convergence of shot-noise fields to stable and Fréchet white noise fields under scaling, introducing the concept of alpha-stable white noise fields.

Findings

Shot-noise fields converge to alpha-stable white noise under scaling.

Extremal shot-noise fields converge to Fréchet white noise.

Results are applied to wireless network analysis.

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 $\alpha$-stable limit, intensity of the underlying point process goes to infinity. It is also shown that the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limte the $\alpha$- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fr\'{e}chet white noise field. Finally, these results are applied to the analysis of wireless networks.