Self-normalized moderate deviations for independent random variables

Abstract

Let X 1,X 2,... be a sequence of independent random variables (r. v. s) belonging to the domain of attraction of a normal or stable law. In this paper, we study moderate deviations for the self-normalized sum Σ i=1 nX i/V n,p where V n,p = (Σ i=1 n{pipe}X i{pipe} p) 1/p (p > 1). Applications to the self-normalized law of the iterated logarithm, Studentized increments of partial sums, t-statistic, and weighted sum of independent and identically distributed (i. i. d.) r. v. s are considered. © 2012 Science China Press and Springer-Verlag Berlin Heidelberg.