Weak Convergence and Its Application

Weak convergence of stochastic processes is one of most important theories in probability theory. Not only probability experts but also more and more statisticians are interested in it. In the study of statistics and econometrics, some problems cannot be solved by the classical method. In this book, we will introduce some recent development of modern weak convergence theory to overcome defects of classical theory.Contents:

The Definition and Basic Properties of Weak Convergence:


Metric Space
The Definition of Weak Convergence of Stochastic Processes and Portmanteau Theorem
How to Verify the Weak Convergence?
Two Examples of Applications of Weak Convergence


Convergence to the Independent Increment Processes:


The Basic Conditions of Convergence to the Gaussian Independent Increment Processes
Donsker Invariance Principle
Convergence of Poisson Point Processes
Two Examples of Applications of Point Process Method


Convergence to Semimartingales:


The Conditions of Tightness for Semimartingale Sequence
Weak Convergence to Semimartingale
Weak Convergence to Stochastic Integral I: The Martingale Convergence Approach
Weak Convergence to Stochastic Integral II: Kurtz and Protter's Approach
Stable Central Limit Theorem for Semimartingales
An Application to Stochastic Differential Equations
Appendix: The Predictable Characteristics of Semimartingales


Convergence of Empirical Processes:


Classical Weak Convergence of Empirical Processes
Weak Convergence of Marked Empirical Processes
Weak Convergence of Function Index Empirical Processes
Weak Convergence of Empirical Processes Involving Time-Dependent data
Two Examples of Applications in Statistics





Readership: Graduate students and researchers in probability & statistics and econometrics.