Weak Convergence and Its Applications

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.
Key Features:

• Summarizes both classical theory and recent development of modern weak convergence theory. Our book is more comprehensive than similar books
• Highlights the martingale convergence method, and give the contrast between this method and other methods
• Presents some applications of weak convergence theory in the fields of statistics and econometrics