On Tempered Discrete and L\'evy White Noises

TL;DR

This paper characterizes when discrete and Lévy white noises are tempered by analyzing their moment properties, providing new insights and a novel proof for the known characterization of tempered Lévy white noises.

Contribution

It offers a new proof and a unified approach connecting discrete and continuous white noises to determine their tempering properties.

Findings

Characterization of tempered discrete white noises.

Characterization of tempered Lévy white noises.

A new proof of the fundamental result on Lévy white noises.

Abstract

We study the growth properties of the family of i.i.d. random sequences, also known as discrete white noises, and of their continuous-domain generalization, the family of L\'evy white noises. More precisely, we characterize the members of both families which are tempered in terms of their moment properties. We recover the characterization of tempered L\'evy white noises obtained by Robert Dalang and Thomas Humeau and provide a new proof of there fundamental result. Our approach is based on a fruitful connection between the discrete and continuous-domain white noises.