# Edgeworth-type expansion in the entropic free CLT

## TL;DR

This paper develops an Edgeworth-type expansion for densities in the free central limit theorem and applies it to the entropic free CLT under a fourth-moment condition, enhancing understanding of convergence rates.

## Contribution

It introduces a new density expansion in the free CLT and extends it to the entropic case with moment assumptions, advancing free probability theory.

## Key findings

- Derived an Edgeworth-type expansion for free CLT densities
- Applied the expansion to the entropic free CLT under fourth-moment conditions
- Provided insights into convergence behavior in free probability

## Abstract

We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order four for the free summands.