Cramer Type Moderate deviations for the Maximum of Self-normalized Sums

Let {X, X-i, i >= 1} be i.i.d. random variables, S-k be the partial sum and V-n(2) = Sigma(n)(i=1) X-i(2). Assume that E(X) = 0 and E(X-4) < infinity. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max(1 <= k <= n) S-k >= x V-n)/(1 - Phi(x)) --> 2 uniformly in x is an element of [0, o(n(1/6)))