A general Darling-Erdös theorem in Euclidean space
arXiv:1608.04549 · doi:10.1007/s10959-016-0728-y
Abstract
We provide an improved version of the Darling-Erdös 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-Erdös theorem.
revised version: some typos found, two proofs with more details + one extra remark about tightness (end of Section 5)