A general Darling-Erd\"os theorem in Euclidean space

TL;DR

This paper extends the Darling-Erd"os theorem to multidimensional random vectors, providing an improved version with a new invariance principle and identifying a borderline case for weak convergence.

Contribution

It introduces a new strong invariance principle for multidimensional sums and refines the Darling-Erd"os theorem in Euclidean space, including a borderline case analysis.

Findings

Extended Darling-Erd"os theorem to multidimensional vectors

Developed a new strong invariance principle in Euclidean space

Identified a borderline case for weak convergence

Abstract

We provide an improved version of the Darling-Erd\"os theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance principle in this setting which has other applications as well such as an integral test refinement of the multidimensional Hartman-Wintner LIL. We also identify a borderline situation where one has weak convergence to a shifted version of the standard limiting distribution in the classic Darling-Erd\"os theorem.