Estimates for the strong approximation in multidimensional central limit theorem

TL;DR

This paper extends strong approximation bounds in the multidimensional central limit theorem, providing explicit dependence on dimension and distribution, generalizing classical results like Komlós–Major–Tusnády.

Contribution

It offers optimal bounds for Gaussian approximation of sums of independent random vectors in multiple dimensions, with explicit constants and generalizations of classical results.

Findings

Explicit bounds depend on dimension and distribution

Generalizes classical strong approximation results

Provides optimal bounds for multidimensional sums

Abstract

In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent $\R^d$-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known results of Koml\'os--Major--Tusn\'ady and Sakhanenko. The dependence of constants on the dimension $d$ and on distributions of summands is given explicitly. Some related problems are discussed.