Local Limit Theorems for Energy Fluxes of Infinite Divisible Random Fields

TL;DR

This paper establishes local limit theorems for divergence-like functionals of infinitely divisible random fields, describing their asymptotic behavior and potential applications in turbulence energy measurement.

Contribution

It derives stable convergence limit theorems for surface integrals of infinitely divisible fields over shrinking regions, linking limits to stochastic integrals with Lévy bases.

Findings

Stable convergence in distribution for divergence functionals

Limit fields described via Lévy basis stochastic integrals

Application to turbulence kinetic energy measurement

Abstract

We study the local asymptotic behavior of divergence-like functionals of a family of $d$-dimensional Infinitely Divisible Random Fields. Specifically, we derive limit theorems of surface integrals over Lipschitz manifolds for this class of fields when the region of integration shrinks to a single point. We show that in most cases, convergence stably in distribution holds after a proper normalization. Furthermore, the limit random fields can be described in terms of stochastic integrals with respect to a L\'evy basis. We additionally discuss how our results can be used to measure the kinetic energy of a possibly turbulent flow.