Weak Convergence of Stochastic Processes: With Applications to Statistical Limit Theorems

by Vidyadhar S. Mandrekar

Overview

The purpose of this book is to present results on the subject of weak convergence in function spaces to study invariance principles in statistical applications to dependent random variables, U-statistics, censor data analysis. Different techniques, formerly available only in a broad range of literature, are for the first time presented here in a self-contained fashion.

Contents:
Weak convergence of stochastic processes
Weak convergence in metric spaces
Weak convergence on C[0, 1] and D[0,∞)
Central limit theorem for semi-martingales and applications
Central limit theorems for dependent random variables
Empirical process
Bibliography

Product Details

ISBN-13:	9783110475456
Publisher:	De Gruyter
Publication date:	09/26/2016
Series:	De Gruyter Textbook , #64
Sold by:	Barnes & Noble
Format:	eBook
Pages:	148
File size:	27 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Vidyadhar Mandrekar, Michigan State University, USA.

Table of Contents

1 Weak convergence of stochastic processes 1

Introduction 1

2 Weak convergence in metric spaces 4

2.1 Cylindrical measures 4

2.2 Kolmogorov consistency theorem 6

2.3 The finite dimensional family for Brownian motion 9

2.4 Properties of Brownian motion 11

2.5 Kolmogorov continuity theorem 14

2.6 Exit time for Brownian motion and Skorokhod theorem 16

2.6.1 Skorokhod theorem 19

2.7 Embedding of sums of i.i.d. random variable in Brownian motion 21

2.8 Donsker's theorem 23

2.9 Empirical distribution function 25

2.10 Weak convergence of probability measure on Polish space 29

2.10.1 Prokhorov theorem 34

2.11 Tightness and compactness in weak convergence 36

3 Weak convergence on C[0,1] and D[0, ∞) 41

3.1 Structure of compact sets in C[0,1] 41

3.1.1 Arzela-Ascoli theorem 41

3.2 Invariance principle of sums of i.i.d. random variables 45

3.3 Invariance principle for sums of stationary sequences 48

3.4 Weak convergence on the Skorokhod space 50

3.4.1 The space D[0,1] 50

3.4.2 Skorokhod topology 52

3.5 Metric of D[0,1] to make it complete 53

3.6 Separability of the Skorokhod space 56

3.7 Tightness in the Skorokhod space 58

3.8 The space D[0, ∞) 60

3.8.1 Separability and completeness 64

3.8.2 Compactness 64

3.8.3 Tightness 66

3.8.4 Aldous's tightness criterion 67

4 Central limit theorem for semi-martingales and applications 70

4.1 Local characteristics of semi-martingale 70

4.2 Lenglart inequality 72

4.3 Central limit theorem for semi-martingale 76

4.4 Application to survival analysis 81

4.5 Asymptotic distribution of β(t) and Kaplan-Meier estimate

5 Central limit theorems for dependent random variables 89

6 Empirical process 110

6.1 Spaces of bounded functions 114

6.2 Maximal inequalities and covering numbers 119

6.3 Sub-Gaussian inequalities 125

6.4 Symmetrization 126

6.4.1 Glivenko-Cantelli theorems 129

6.4.2 Donsker theorems 131

6.5 Lindberg-type theorem and its applications 133

Bibliography 142

Editorial Reviews

"Written by an expert in probability theory and stochastic processes, the book succeeds to present, in a relatively small number of pages, some fundamental results on weak convergence in probability theory and stochastic process and applications."
Hannelore Lisei in: Stud. Univ. Babes-Bolyai Math. 62(2017), No. 1, 137-138

From the Publisher