A general Darling-Erd\"os theorem in Euclidean space
Probability
2016-12-05 v2
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.
Cite
@article{arxiv.1608.04549,
title = {A general Darling-Erd\"os theorem in Euclidean space},
author = {Gauthier Dierickx and Uwe Einmahl},
journal= {arXiv preprint arXiv:1608.04549},
year = {2016}
}
Comments
revised version: some typos found, two proofs with more details + one extra remark about tightness (end of Section 5)