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lnotes - Probability Theory STAT310/MATH230 May 1 2010 Amir...

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Unformatted text preview: Probability Theory: STAT310/MATH230; May 1, 2010 Amir Dembo E-mail address : [email protected] Department of Mathematics, Stanford University, Stanford, CA 94305. Contents Preface 5 Chapter 1. Probability, measure and integration 7 1.1. Probability spaces, measures and σ-algebras 7 1.2. Random variables and their distribution 17 1.3. Integration and the (mathematical) expectation 30 1.4. Independence and product measures 53 Chapter 2. Asymptotics: the law of large numbers 69 2.1. Weak laws of large numbers 69 2.2. The Borel-Cantelli lemmas 75 2.3. Strong law of large numbers 83 Chapter 3. Weak convergence, clt and Poisson approximation 93 3.1. The Central Limit Theorem 93 3.2. Weak convergence 101 3.3. Characteristic functions 115 3.4. Poisson approximation and the Poisson process 127 3.5. Random vectors and the multivariate clt 135 Chapter 4. Conditional expectations and probabilities 145 4.1. Conditional expectation: existence and uniqueness 145 4.2. Properties of the conditional expectation 150 4.3. The conditional expectation as an orthogonal projection 158 4.4. Regular conditional probability distributions 162 Chapter 5. Discrete time martingales and stopping times 167 5.1. Definitions and closure properties 167 5.2. Martingale representations and inequalities 176 5.3. The convergence of Martingales 182 5.4. The optional stopping theorem 195 5.5. Reversed MGs, likelihood ratios and branching processes 200 Chapter 6. Markov chains 215 6.1. Canonical construction and the strong Markov property 215 6.2. Markov chains with countable state space 223 6.3. General state space: Doeblin and Harris chains 244 Chapter 7. Continuous, Gaussian and stationary processes 259 7.1. Definition, canonical construction and law 259 7.2. Continuous and separable modifications 264 3 4 CONTENTS 7.3. Gaussian and stationary processes 274 Chapter 8. Continuous time martingales and Markov processes 279 8.1. Continuous time filtrations and stopping times 279 8.2. Continuous time martingales 284 8.3. Markov and Strong Markov processes 307 Chapter 9. The Brownian motion 323 9.1. Brownian transformations, hitting times and maxima 323 9.2. Weak convergence and invariance principles 330 9.3. Brownian path: regularity, local maxima and level sets 348 Bibliography 355 Index 357 310c: Homework Solutions 2010 363 Preface These are the lecture notes for a yearly PhD level course in Probability Theory that I taught at Stanford University in 2004, 2006 and 2009. The goal of this course is to prepare incoming PhD students in Stanford’s mathematics and statistics departments to do research in probability theory. More broadly, the goal of the text is to help the reader master the mathematical foundations of probability theory and the techniques most commonly used in proving theorems in this area. This is then applied to the rigorous study of the most fundamental classes of stochastic processes....
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