Elliptical Tempered Stable Distribution and Fractional Calculus

TL;DR

This paper introduces the elliptical tempered stable distribution, linking it to fractional calculus, and provides analytical approximations for its probability density function, expanding the understanding of tempered infinite divisible distributions.

Contribution

It defines the elliptical tempered stable distribution via characteristic functions and connects it with fractional calculus, offering new analytical tools.

Findings

Defined elliptical tempered stable distribution using spectral measure

Established a connection with fractional calculus

Provided analytical approximations for the distribution's density

Abstract

A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, with fractional calculus. Finally, some analytical approximations for the probability density function of tempered infinite divisible distribution, which elliptical tempered stable distributions are a subclass of them, are considered.