A direct approach to the stable distributions

TL;DR

This paper derives an explicit form for the characteristic function of stable distributions using analytical methods, avoiding the traditional Lévy-Khintchine approach, thus providing a new perspective on their mathematical structure.

Contribution

It introduces a direct analytical method to obtain the characteristic function of stable distributions, bypassing the Lévy-Khintchine integral representation.

Findings

Explicit characteristic function derived analytically

Avoids reliance on Lévy-Khintchine representation

Provides new insights into stable distribution structure

Abstract

The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying theory of regular variation, without appeal to the general L\'evy-Khintchine integral representation of infinitely divisible distributions.