L\'evy Processes and L\'evy White Noise as Tempered Distributions

TL;DR

This paper establishes a precise criterion based on the Le9vy measure's moments for when Le9vy white noise can be considered a tempered distribution, clarifying the conditions under which such noise belongs to this functional space.

Contribution

It provides a necessary and sufficient condition linking the moments of the Le9vy measure to the distributional properties of the associated white noise.

Findings

Le9vy white noise is a tempered distribution if the Le9vy measure has a positive absolute moment.

If the Le9vy measure lacks positive absolute moments, the white noise is almost surely not a tempered distribution.

The paper characterizes the probabilistic behavior of Le9vy white noise in relation to tempered distributions.

Abstract

We identify a necessary and sufficient condition for a L\'evy white noise to be a tempered distribution. More precisely, we show that if the L\'evy measure associated with this noise has a positive absolute moment, then the L\'evy white noise almost surely takes values in the space of tempered distributions. If the L\'evy measure does not have a positive absolute moment of any order, then the event on which the L\'evy white noise is a tempered distribution has probability zero.