Almost sure versions of the Darling-Erdős theorem

We prove a.s. limit theorems corresponding to the classical Darling-Erdős theorem for the maxima of normalized partial sums of i.i.d. random variables. Our results yield the analogue of the a.s. central limit theorem for the Darling-Erdős max functional and its variants. Unlike in standard a.s. central limit theory, our theorems involve nonlogarithmic averages.