ScholtzChuggNotes

ScholtzChuggNotes - Supplemental Notes on Random Processes...

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Unformatted text preview: Supplemental Notes on Random Processes (A Work in Progress) ( marks incomplete sections) Robert Scholtz and Keith Chugg January 3, 2007 ii Contents 1 A Roadmap 1 1.1 The Big Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Starting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Notes on Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Appendix: Preparedness Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Preliminaries I: Computations with Random Variables 9 2.1 The Expected Value Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Evaluation of Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 The Validity of the Function Approach to Expectation Calculations . . . . . 12 2.1.3 The Calculus of Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.4 Complex Random Variables and the Expected-Value Operator . . . . . . . . 15 2.2 A Review of Transformations of Random Variables . . . . . . . . . . . . . . . . . . . 17 2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Preliminaries II: Spaces and Convergence 27 3.1 Linear Spaces and Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Metric Spaces and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1 The Properties of Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Useful Inequalities Involving Random Variables . . . . . . . . . . . . . . . . . 35 3.2.3 Convergence of Sequences of Random Variables in Metric Spaces . . . . . . . 40 3.2.3.1 Almost Sure Convergence . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3.2 Convergence in the r th Mean . . . . . . . . . . . . . . . . . . . . . . 42 3.2.3.3 Convergence in Probability . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.4 Convergence of Sequences of Distribution Functions . . . . . . . . . . . . . . 45 3.2.5 Convergence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Combined Algebraic and Topological properties . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 Norms, Banach Spaces, and Operators . . . . . . . . . . . . . . . . . . . . . . 50 3.3.2 Inner Products and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.3 The Hilbert Space L 2 ( U , F , P ) of Random Variables . . . . . . . . . . . . . . 57 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 Preliminaries III: Characteristic Functions 63 4.1 Characteristic Functions of Random Variables . . . . . . . . . . . . . . . . . . . . . . 63 iii iv CONTENTS 4.1.1 Non-negative Denite Functions . . . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 05/06/2008 for the course EE 562a taught by Professor Toddbrun during the Spring '07 term at USC.

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ScholtzChuggNotes - Supplemental Notes on Random Processes...

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