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

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.

26 pages

References in corpus (3)

• Normal approximation under local dependence
• Normal approximation for nonlinear statistics using a concentration inequality approach
• Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs