Nonasymptotic and distribution-uniform Koml\'os-Major-Tusn\'ady approximation

TL;DR

This paper develops nonasymptotic, distribution-uniform Gaussian approximation bounds for sums of i.i.d. random variables, extending KMT results with explicit constants and conditions for uniform convergence rates.

Contribution

It provides the first nonasymptotic, distribution-uniform KMT approximation with explicit constants and necessary and sufficient conditions for uniform convergence rates.

Findings

Universal or explicit constants in inequalities.

Distribution-uniform generalizations of KMT approximations.

Necessary and sufficient conditions for uniform convergence rates.

Abstract

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Koml\'os, Major, and Tusn\'ady (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.