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Davar Khoshnevisan Graduate Studies in Mathematics Volume 80 American Mathematical Society Probability Probability Davar Khoshnevisan Graduate Studies in Mathematics Volume 80 American Mathematical Society Providence, Rhode Island Editorial Board David Cox Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 60- 01; Secondary 60-03, 28-01, 28-03. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-80 Library of Congress Cataloging-in-Publication Data Khoshnevisan, Davar. Probability / Davar Khoshnevisan. p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 80) Includes bibliographical references and index. ISBN-13: 978-0-8218-4215-7 (alk. paper) ISBN-10: 0-8218-4215-3 (alk. paper) 1. Probabilities. 1. Title. QA273.K488 2007 2006052603 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also he made by e-mail to reprint-peraissionaams.org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://ww.am8.org/ 10987654321 1211 10090807 To my family Contents Preface ................................... xi General Notation ..... ......................... xv Classical Probability ..................... Chapter 1. 1 Discrete Probability ........................ §1. 1 . .... . .. .. ...... . .. §2. Conditional Probability . . . 4 Independence ........... . ....... ...... . §3. .. .. ... . ..... ... 6 Discrete Distributions ... . .... . §4. 6 .. .. .. .. §5. Absolutely Continuous Distributions .. . . . . . 10 Expectation and Variance ..................... 12 §6. . .... ... ......... . .... . Problems .... .... . 13 Notes .......... .......................... 15 . .... .. . ..... ....... Chapter 2. Bernoulli Trials . . . 17 The Classical Theorems ...................... 18 §1. . ..... . . .... .. Problems .... ...... .. Notes .................. ........... ............ 2. 221 .. .. . ... . .. Chapter 3. Measure Theory . . . . . . ..... .... . ....... 23 Measure Spaces .... .. ..... .. .. §1. 23 . ... . .. .. ..... . §2. Lebesgue Measure .... . . . . . . 25 . .... . . .... . ... . .. §3. Completion . . . . . . . . ........ ...... 28 §4. Proof o.f .C.a.r.a t.heo..dory's T.h e.o.r.e.m. ...... ... . .... . .. 30 Problems . . . . . . . 33 vii viii Contents Notes ................................... 34 . .... . . .... . ... . .. .. .. Chapter 4. Integration . . . . 35 Measurable Functions ....................... 35 §1. . ... . ... ... .. .. §2. The Abstract Integral ..... .. .. ..... . . .... 37 §3. LP-Spaces ..... .. .... ... . . .. ....... ... . ... 39 Modes of Convergence .... ... . §4. . ......... ... ....... . . ..... 454:3 §5. Limit Theorems §6. The Radon-Nikodym Theorem .. ......... .. .. . . . 47 Problems ........ ... . ... .. .......... . ..... 49 Notes ................................... 52 Product Spaces ........................ 53 Chapter 5. . ....... .. ........... ..... §1. Finite Products 53 Infinite Products ....... . ........... . . ..... 58 §2. §P3.roCobmplleemment:s P ro.o.f. o.f. K..ol.m.o.g.o.r.o.v.'s. E.x.t.e.n.si.o.n. T.h.e.o.re.m...... 6. 260 Notes ................................... 64 .. .. .. . ...... .. Chapter 6. Independence ..... . . . . . 65 Random Variables and Distributions .... . .... . .... . §1. 65 Independent Random Variables .. ...... ....... .. §2. . ............ . ...... 67 §3. An Instructive Example .. . 71 §4. Khintchine's Weak Law of Large Numbers .... .. ..... 71 Kolmogorov's Strong Law of Large Numbers ..... ..... 73 §5. .. ...... . ............. ..... §6. Applications . .. ..... . . ........... .. ..... 77 Problems .... . 84 Notes ..... ............................... 89 . ... .. .. .. .... . Chapter 7. The Central Limit Theore.m ... . .... . ...... . . 91 §1. Weak Convergence ..... . . . 91 . ... §2. Weak Convergence and Compact-Support Functions .. 94 Harmonic Analysis in Dimension One .... .. ....... . §3. .. .. ...... .. . .... ... 96 §4. The Plancherel Theorem . . ....... . 97 . ... 100 §5. The 1-D Central Limit Theor.e.m ......... ........ ..... 101 §6. Complements to the CLT . Problems ........ ....... .. .......... . .. ... 111 Notes .......... ....... .. .. ....... . . ... 117 . . Contents ix Martingales .. ............ ..... . ...... 119 Chapter 8. . .... . ... ........... . §1. Conditional Expectations 119 Filtrations and Semi-Martingales ................. 126 §2. . ..... ..... . §3. Stopping Times and Optional Stopping . 129 Applications to Random Walks .................. 131 §4. . ...... . . .... . §5. Inequalities and Convergence .... . 134 Further Applications ........................ 136 §6. Problems ....................... . . ... . .... ........... ....... .. ....... .. ...... 151 Notes 157 Brownian Motion ...................... 159 Chapter 9. Gaussian Processes ..... .... ... ...... . .... . §1. 160 . ..... 165 §2. Wiener's Construction: Brownian Motion on [0.1) . Nowhere-Differentiability ..................... 168 §3. §4. The Brownian Filtration and Stopping Times .... . .... . 170 The Strong Markov Property .. .. ... ...... . ..... 17.3 §5. The Reflection Principle ... .... ....... .. ...... 175 §6. Problems ................................. 176 . ....... . ...... .. ......... ..... . Notes .. 180 . . .... . .... . Chapter 10. Terminus: Stochastic I.n.te.g.r.a ti.o.n. . ... .. ... .. 181 §1. The Indefinite Ito Integral . . . 181 Continuous Martingales in LZ(P) .. ... ...... ...... 187 §2. The Definite Ito Integral ......... ............ 189 §3. . ...... ... Quadratic Variation ... ..... .... .. §4. 192 §5. Ito's Formula and Two Applicati.o n.s. ..... .... ........... ..... 193 Problems ................... 199 . ....... ... Notes ..... .. .... ... .... . .... . 201 Appendix ........................ .......... 203 Hilbert Spaces ........................... 203 §1. Fourier Series .................. ....... . §2. . . 205 Bibliography ...................... .......... 209 ........ .......... ...... . . .... . .. Index . . . 217

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This is a textbook for a one-semester graduate course in measure-theoretic probability theory, but with ample material to cover an ordinary year-long course at a more leisurely pace. Khoshnevisan's approach is to develop the ideas that are absolutely central to modern probability theory, and to show
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Davar Khoshnevisan Graduate Studies in Mathematics Volume 80 American Mathematical Society Probability Probability Davar Khoshnevisan Graduate Studies in Mathematics Volume 80 American Mathematical Society Providence, Rhode Island Editorial Board David Cox Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 60- 01; Secondary 60-03, 28-01, 28-03. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-80 Library of Congress Cataloging-in-Publication Data Khoshnevisan, Davar. Probability / Davar Khoshnevisan. p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 80) Includes bibliographical references and index. ISBN-13: 978-0-8218-4215-7 (alk. paper) ISBN-10: 0-8218-4215-3 (alk. paper) 1. Probabilities. 1. Title. QA273.K488 2007 2006052603 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also he made by e-mail to reprint-peraissionaams.org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://ww.am8.org/ 10987654321 1211 10090807 To my family Contents Preface ................................... xi General Notation ..... ......................... xv Classical Probability ..................... Chapter 1. 1 Discrete Probability ........................ §1. 1 . .... . .. .. ...... . .. §2. Conditional Probability . . . 4 Independence ........... . ....... ...... . §3. .. .. ... . ..... ... 6 Discrete Distributions ... . .... . §4. 6 .. .. .. .. §5. Absolutely Continuous Distributions .. . . . . . 10 Expectation and Variance ..................... 12 §6. . .... ... ......... . .... . Problems .... .... . 13 Notes .......... .......................... 15 . .... .. . ..... ....... Chapter 2. Bernoulli Trials . . . 17 The Classical Theorems ...................... 18 §1. . ..... . . .... .. Problems .... ...... .. Notes .................. ........... ............ 2. 221 .. .. . ... . .. Chapter 3. Measure Theory . . . . . . ..... .... . ....... 23 Measure Spaces .... .. ..... .. .. §1. 23 . ... . .. .. ..... . §2. Lebesgue Measure .... . . . . . . 25 . .... . . .... . ... . .. §3. Completion . . . . . . . . ........ ...... 28 §4. Proof o.f .C.a.r.a t.heo..dory's T.h e.o.r.e.m. ...... ... . .... . .. 30 Problems . . . . . . . 33 vii viii Contents Notes ................................... 34 . .... . . .... . ... . .. .. .. Chapter 4. Integration . . . . 35 Measurable Functions ....................... 35 §1. . ... . ... ... .. .. §2. The Abstract Integral ..... .. .. ..... . . .... 37 §3. LP-Spaces ..... .. .... ... . . .. ....... ... . ... 39 Modes of Convergence .... ... . §4. . ......... ... ....... . . ..... 454:3 §5. Limit Theorems §6. The Radon-Nikodym Theorem .. ......... .. .. . . . 47 Problems ........ ... . ... .. .......... . ..... 49 Notes ................................... 52 Product Spaces ........................ 53 Chapter 5. . ....... .. ........... ..... §1. Finite Products 53 Infinite Products ....... . ........... . . ..... 58 §2. §P3.roCobmplleemment:s P ro.o.f. o.f. K..ol.m.o.g.o.r.o.v.'s. E.x.t.e.n.si.o.n. T.h.e.o.re.m...... 6. 260 Notes ................................... 64 .. .. .. . ...... .. Chapter 6. Independence ..... . . . . . 65 Random Variables and Distributions .... . .... . .... . §1. 65 Independent Random Variables .. ...... ....... .. §2. . ............ . ...... 67 §3. An Instructive Example .. . 71 §4. Khintchine's Weak Law of Large Numbers .... .. ..... 71 Kolmogorov's Strong Law of Large Numbers ..... ..... 73 §5. .. ...... . ............. ..... §6. Applications . .. ..... . . ........... .. ..... 77 Problems .... . 84 Notes ..... ............................... 89 . ... .. .. .. .... . Chapter 7. The Central Limit Theore.m ... . .... . ...... . . 91 §1. Weak Convergence ..... . . . 91 . ... §2. Weak Convergence and Compact-Support Functions .. 94 Harmonic Analysis in Dimension One .... .. ....... . §3. .. .. ...... .. . .... ... 96 §4. The Plancherel Theorem . . ....... . 97 . ... 100 §5. The 1-D Central Limit Theor.e.m ......... ........ ..... 101 §6. Complements to the CLT . Problems ........ ....... .. .......... . .. ... 111 Notes .......... ....... .. .. ....... . . ... 117 . . Contents ix Martingales .. ............ ..... . ...... 119 Chapter 8. . .... . ... ........... . §1. Conditional Expectations 119 Filtrations and Semi-Martingales ................. 126 §2. . ..... ..... . §3. Stopping Times and Optional Stopping . 129 Applications to Random Walks .................. 131 §4. . ...... . . .... . §5. Inequalities and Convergence .... . 134 Further Applications ........................ 136 §6. Problems ....................... . . ... . .... ........... ....... .. ....... .. ...... 151 Notes 157 Brownian Motion ...................... 159 Chapter 9. Gaussian Processes ..... .... ... ...... . .... . §1. 160 . ..... 165 §2. Wiener's Construction: Brownian Motion on [0.1) . Nowhere-Differentiability ..................... 168 §3. §4. The Brownian Filtration and Stopping Times .... . .... . 170 The Strong Markov Property .. .. ... ...... . ..... 17.3 §5. The Reflection Principle ... .... ....... .. ...... 175 §6. Problems ................................. 176 . ....... . ...... .. ......... ..... . Notes .. 180 . . .... . .... . Chapter 10. Terminus: Stochastic I.n.te.g.r.a ti.o.n. . ... .. ... .. 181 §1. The Indefinite Ito Integral . . . 181 Continuous Martingales in LZ(P) .. ... ...... ...... 187 §2. The Definite Ito Integral ......... ............ 189 §3. . ...... ... Quadratic Variation ... ..... .... .. §4. 192 §5. Ito's Formula and Two Applicati.o n.s. ..... .... ........... ..... 193 Problems ................... 199 . ....... ... Notes ..... .. .... ... .... . .... . 201 Appendix ........................ .......... 203 Hilbert Spaces ........................... 203 §1. Fourier Series .................. ....... . §2. . . 205 Bibliography ...................... .......... 209 ........ .......... ...... . . .... . .. Index . . . 217

Description:
This is a textbook for a one-semester graduate course in measure-theoretic probability theory, but with ample material to cover an ordinary year-long course at a more leisurely pace. Khoshnevisan's approach is to develop the ideas that are absolutely central to modern probability theory, and to show
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