A General Darling–Erdős Theorem in Euclidean Space

We provide an improved version of the Darling–Erdős 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 classical Darling–Erdős theorem.