Darling-Erdős Theorem for Self-Normalized Sums

Let X,X1,X2,... be i.i.d. nondegenerate random variables, Sn=∑j=1 nXjand Vn 2=∑j=1 nXj 2. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤ k≤ nSk/Vk, and the law of the iterated logarithm for self-normalized sums, Sn/Vn, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling-Erdős-type theorem as well as an Erdős-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.