# On the error bound in the normal approximation for Jack measures

## TL;DR

This paper establishes bounds on the accuracy of normal approximation for Jack measures using Stein's method, advancing understanding of character ratios and zero-bias couplings.

## Contribution

It provides near-optimal uniform bounds and introduces a Rosenthal-type inequality for zero-bias couplings in the context of Jack measures.

## Key findings

- Uniform bounds close to conjectured limits
- Non-uniform bounds for character ratios
- New Rosenthal-type inequality for zero-bias couplings

## Abstract

In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein's method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman [J. Combin. Theory Ser. A, 108 (2004), 275--296]. As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.