  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.
