Darling-Erdos theorem for self-normalized sums

Abstract

Let X, X1, X2,... be i.i.d. nondegenerate random variables, Sn = ∑j=1n Xj and Vn2 = ∑j=1n. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤k≤n Sk/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-Erdos-type theorem as well as an Erdos-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.