# Rates of convergence in the CLT for nonlinear statistics under relaxed moment conditions

## TL;DR

This paper establishes explicit convergence rates in the central limit theorem for nonlinear statistics under relaxed moment conditions using Stein's method, extending previous results to broader moment assumptions.

## Contribution

It provides new explicit rates of convergence for nonlinear statistics with relaxed moment conditions, including cases with vanishing third moments.

## Key findings

- Rates of convergence are of optimal order $O(n^{-rac{ ext{ extdelta}}{2}})$ and $O(n^{-rac{1+ extdelta}{2}})$.
- Results apply to nonlinear statistics with finite moments of order $2+ extdelta$ and $3+ extdelta$.
- Method uses covariance identities and solutions to Stein's equation.

## Abstract

This paper is concerned with normal approximation under relaxed moment conditions using Stein's method. We obtain the explicit rates of convergence in the central limit theorem for (i) nonlinear statistics with finite absolute moment of order $2+\delta\in(2,3];$ (ii) nonlinear statistics with vanishing third moment and finite absolute moment of order $3+\delta\in(3,4].$ When applied to specific examples, these rates are of the optimal order $O(n^{-\frac{\delta}{2}})$ and $O(n^{-\frac{1+\delta}{2}}).$ Our proof are based on the covariance identify formula and simple observations about the solution of Stein's equation.