Stochastic integral and series representations for strictly stable distributions

TL;DR

This paper develops stochastic integral representations for strictly stable distributions, showing that these representations encompass a broader class than shot-noise series, especially when the stability index exceeds 1.

Contribution

It establishes a new stochastic integral representation for strictly stable distributions and clarifies the relationship with shot-noise series representations, including explicit descriptions of distributions with both.

Findings

Stochastic integral representation includes more distributions than shot-noise series.

Proper inclusion of classes when the stability index > 1.

Explicit characterization of distributions with both representations.

Abstract

In this paper we find and develop a stochastic integral representation for the class of strictly stable distributions. We establish an explicit relationship between stochastic integral and shot-noise series representations of strictly stable distributions, which shows that the class of distributions representable by stochastic integral is larger than the class representable by a shot-noise series. This inclusion is proper when the stability index is greater than 1. We also give an explicit description of distributions possessing both representations.