# Normal Approximation for Weighted Sums under a Second Order Correlation Condition

## TL;DR

This paper establishes a bound on how closely weighted sums of dependent variables approximate a normal distribution under certain correlation conditions, using advanced concentration inequalities, with applications to log-concave measures.

## Contribution

It introduces a new upper bound for the normal approximation of dependent weighted sums under second order correlation conditions, improving existing results.

## Key findings

- Bound of order (log n)/n for Kolmogorov distance
- Enhanced concentration inequalities on high-dimensional spheres
- Applications to log-concave probability measures

## Abstract

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.