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

2019/05/18 by Nguyen Tien Dung, Dung, Nguyen Tien
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1905.07564

26 pages

arxiv created 2021/06/15 · arxiv updated 2021/06/16

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+δ∈(2,3]; (ii) nonlinear statistics with vanishing third moment and finite absolute moment of order 3+δ∈(3,4]. When applied to specific examples, these rates are of the optimal order O(n^-\fracδ2) and O(n^-(1+δ)/(2)). Our proof are based on the covariance identify formula and simple observations about the solution of Stein's equation.

Related

• Explicit rates of convergence in the multivariate CLT for nonlinear statistics
• Stein kernels for q-moment measures and new bounds for the rate of convergence in the central limit theorem
• Rates of convergence in the central limit theorem for Banach valued dependent variables
• Quantitative Normal Approximation of Linear Statistics of β-ensembles
• A Bound on the Rate of Convergence in the Central Limit Theorem for Renewal Processes under Second Moment Conditions