Extension of the Darling and Erdos Theorem on the Maximum of Normalized Sums

Abstract

The limiting distribution of $\max_{1\leqslant k \leqslant n}S_k/k^{\frac{1}{2}}$ is derived via embedding. The theorem also applies to partial sums of certain dependent rv's. Thus the proof of the Darling and Erdos result is brought into line with recent literature; moreover, the scope of its applicability is greatly increased in regard both to dependence and moment assumptions.