Fractional Edgeworth Expansion: Corrections to the Gaussian-L\'evy Central Limit Theorem

TL;DR

This paper extends the classical Edgeworth expansion to sums of variables with infinite variance, providing a series correction involving fractional derivatives to improve the approximation of their probability density functions.

Contribution

It introduces a fractional Edgeworth expansion applicable to L\'evy-stable distributions, including the Gaussian case, with new special functions and practical implementation for atomic momentum distributions.

Findings

Improved approximation of L\'evy-stable distributions

Introduction of new special functions for series expansion

Enhanced accuracy near the L\'evy-Gaussian transition

Abstract

In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to sums of variables with an infinite variance which converge by the generalized central limit theorem to a L\'evy $\alpha$-stable density function. Our correction may be written by means of a series of fractional derivatives of the L\'evy and the conjugate L\'evy PDFs. This series expansion is general and applies also to the Gaussian regime. To describe the terms in the series expansion, we introduce a new family of special functions and briefly discuss their properties. We implement our generalization to the distribution of the momentum for atoms undergoing Sisyphus cooling, and show the improvement of our leading order approximation compared to previous approximations. In vicinity of the transition between L\'{e}vy and Gauss behaviors, convergence to asymptotic results slows down.