Limit theorems in probability

Limit theorems in probability - Limit theorems in...

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Unformatted text preview: Limit theorems in probability Dr. Bing-Yi JING Dept. of Math., HKUST Email: majing@ust.hk September 15, 2011 Contents 1 Set Theory 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Operations of sequence of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Indicator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Semi-algebras, Algebras, and σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.2 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.3 Some special σ-algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.4 How to generate algebras from semi-algebras . . . . . . . . . . . . . . . . . . . . . 7 1.6 Generated classes (Minimal classes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 Monotone class ( m-class), π-class, and λ-class . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7.2 Relationships with σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7.3 Minimal m-class, λ-class and π-class . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7.4 Graphical illustration of different classes . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 The Monotone Class Theorem (MCT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Measure Theory 20 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Properties of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Case I: semialgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Case II: algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Case III: σ-algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Arithmetics with ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Probability measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Some examples of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Extension of set functions (or measures) from semialgebras to algebras . . . . . . . . . . . 25 2.7 Outer measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2....
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This note was uploaded on 02/01/2012 for the course MATH 5010 taught by Professor D during the Spring '11 term at HKU.

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