Weak convergence and its applications / Zhengyan Lin, Hanchao Wang.Material type: TextPublication details: Singapore ; Hackensack, N.J. : World Scientific Pub. Co., ©2014. Description: 1 online resource (viii, 176 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789814447706; 9814447706Subject(s): Convergence | Distribution (Probability theory) | MATHEMATICS -- Applied | MATHEMATICS -- Probability & Statistics -- General | Convergence | Distribution (Probability theory)Genre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: No titleDDC classification: 519.2 LOC classification: QA274Online resources: Click here to access online
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Includes bibliographical references (pages 171-174) and index.
1. The definition and basic properties of weak convergence. 1.1. Metric space. 1.2. The definition of weak convergence of stochastic processes and portmanteau theorem. 1.3. How to verify the weak convergence? 1.4. Two examples of applications of weak convergence -- 2. Convergence to the independent increment processes. 2.1. The basic conditions of convergence to the Gaussian independent increment processes. 2.2. Donsker invariance principle. 2.3. Convergence of Poisson point processes. 2.4. Two examples of applications of point process method -- 3. Convergence to semimartingales. 3.1. The conditions of tightness for semimartingale sequence. 3.2. Weak convergence to semimartingale. 3.3. Weak convergence to stochastic integral I: the martingale convergence approach. 3.4. Weak convergence to stochastic integral II: Kurtz and Protter's approach. 3.5. Stable central limit theorem for semimartingales. 3.6. An application to stochastic differential equations -- 4. Convergence of empirical processes. 4.1. Classical weak convergence of empirical processes. 4.2. Weak convergence of marked empirical processes. 4.3. Weak convergence of function index empirical processes. 4.4. Weak convergence of empirical processes involving time-dependent data. 4.5. Two examples of applications in statistics.
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.