Strong Limit Theorems For Self-normalized Partial Sums Of Random Variables

This dissertation focuses on the strong limit theorems for self-normalized partial sums of i.i.d. or weakly dependent random variables and for the rescaled range statistic, which is defined by a self-normalized form.This thesis consists of five chapters as in the following features.In Chapter1, we introduce the background and some auxiliary information of the research work carried out in this thesis.In Chapter2, we give an almost sure central limit theorem for self-normalized partial sums of a strictly stationary φ-mixing sequences which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. It substantially extends a result on the almost sure central limit theorem previously obtained by Huang and Pang (2010).In Chapter3, we consider an almost sure central limit theorem for self-normalized products of sum of partial sums for a sequence of strictly stationary φ-mixing positive random variables which are in the domain of attraction of the normal law with positive mean, possibly infinite variance holds under a fairly general growth condition on the weight sequence.In Chapter4, we establish two precise asymptotics related to probability convergence for the rescaled range statistic. Moreover, a precise asymptotics related to almost surely convergence for the rescaled range statistic is also considered under some mild conditions.In final Chapter5, we show a nonclassical law of iterated logarithm for self-normalized partial sums of a sequence of nondegenerate, symmetric, i.i.d. random variables which are in the domain of attraction of the normal law with zero means and possibly infinite variances.