Darling--Erdős theorem for self-normalized sums

Abstract

Let $X,\, X_1,\, X_2,\ldots$ be i.i.d. nondegenerate random variables, $S_n=\sum_{j=1}^nX_j$ and $V_n^2=\sum_{j=1}^nX_j^2$. We investigate the asymptotic \vspace*{1pt} behavior in distribution of the maximum of self-normalized sums, $\max_{1\le k\le n}S_k/V_k$, and the law of the iterated logarithm for self-normalized sums, $S_n/V_n$, 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.