Strong limit theorems for self-normalized partial sums of random variables

摘要

This dissertation focuses on the strong limit theorems for self-normalized partialsums 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 Chapter 1, we introduce the background and some auxiliary information of theresearch work carried out in this thesis.
　　In Chapter 2, we give an almost sure central limit theorem for self-normalized partialsums of a strictly stationary φ-mixing sequences which is in the domain of attraction ofthe normal law with mean zero and possibly infinite variance.It substantially extendsa result on the almost sure central limit theorem previously obtained by Huang andPang (2010).
　　In Chapter 3, we consider an almost sure central limit theorem for self-normalizedproducts of sum of partial sums for a sequence of strictly stationary φ-mixing positiverandom variables which are in the domain of attraction of the normal law with positivemean, possibly infinite variance holds under a fairly general growth condition on theweight sequence.
　　In Chapter 4, we establish two precise asymptotics related to probability convergencefor the rescaled range statistic.Moreover, a precise asymptotics related to almostsurely convergence for the rescaled range statistic is also considered under some mildconditions.
　　In final Chapter 5, we show a nonclassical law of iterated logarithm for self-normalizedpartial sums of a sequence of nondegenerate, symmetric, i.i.d.random variables whichare in the domain of attraction of the normal law with zero means and possibly infinitevariances.
　　Keyword: almost sure central limit theorem; φ-mixing; domain of attraction of thenormal law; self-normalized partial sum; strictly stationary; self-normalized product ofsum of partial sums; law of the iterated logarithm; precise asymptotics; rescaled rangestatistic; nonclassical law of the iterated logarithm

目录

声明

Acknowledgements

Preface

Abstract

Contents

Chapter 1 Preliminaries

1．1 ASCLT for partial sums

1．2 ASCLT for products of sum of partial sums

1．3 The weakly dependent random variables

1．4 The self-normalized partial sums

1．5 The precise asymptotics

1．6 The rescaled range statistic

1．7 The nonclassical law of the iterated logarithm

1．8 Notation

Chapter 2 ASCLT for self-normalized partial sums

2．1 Introduction and main results

2．2 Proofs

Chapter 3 ASCLT for self-normalized products of sum of partial sums

3．1 Introduction and main results

3．2 Lemmas

3．3 Proof of Theorem 3．1．1．

Chapter 4 Precise asymptotics in the LIL for R／S statistic

4．1 Introduction and main results

4．2 Normal case

4．3 Truncation and Approximation

4．4 Proofs of Theorems 4．1．1 and 4．1．2

4．5 Proof of Theorem 4．1．3

Chapter 5 A nonclassical LIL for self-normalized partial sums

5．1 Introduction and main result

5．2 Proofs

Bibliography

List of papers