Asymptotic Expansions and two-sided Bounds in Randomized Central Limit Theorems

TL;DR

This paper investigates bounds on the distance between distributions of weighted sums of dependent variables and the normal distribution, providing asymptotic expansions and illustrating results for variables on Euclidean spheres.

Contribution

It introduces new bounds and asymptotic expansions for dependent sums, and surveys improved normal approximation rates in randomized CLTs.

Findings

Bounds for Kolmogorov and L^2 distances are established.

Results are illustrated for variables supported on Euclidean spheres.

The paper surveys improved rates of normal approximation in randomized CLTs.

Abstract

Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables whose joint distributions are supported on Euclidean spheres. We also survey several results on improved rates of normal approximation in randomized central limit theorems.