Two multivariate central limit theorems

TL;DR

This paper establishes explicit error bounds for approximating projections of certain high-dimensional random vectors by Gaussian vectors, extending classical CLT results to more general dependence structures.

Contribution

It provides new bounds for rank $k$ projections of random vectors with independent but non-identically distributed components and exchangeable sequences.

Findings

Derived explicit error bounds for Gaussian approximation

Extended CLT to non-i.i.d. and exchangeable cases

Applicable to high-dimensional data projections

Abstract

In this paper, explicit error bounds are derived in the approximation of rank $k$ projections of certain $n$-dimensional random vectors by standard $k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$, $n$, and a basis of the $k$-dimensional space onto which we project. The random vectors considered are two generalizations of the case of a vector with independent, identically distributed components. In the first case, the random vector has components which are independent but need not have the same distribution. The second case deals with finite exchangeable sequences of random variables.