Probability: Theory and Examples Rick Durrett Version 5 ...

1 IProbability: Theory and ExamplesRick DurrettVersion 5 January 11, 2019 Copyright 2019, All rights times the lights are shining on me. Other times I canbarely it occurs to me what a long strange trip its DeadIn 1989 when the first edition of the book was completed, my sonsDavid and Greg were 3 and 1, and the cover picture showed the DowJones at 2650. The last twenty-nine years have brought many changesbut the song remains the same. The title of the book indicates that as wedevelop the Theory , we will focus our attention on Examples .

2 Hoping thatthe book would be a useful reference for people who apply probabilityin their work, we have tried to emphasize the results that are importantfor applications, and illustrated their use with roughly 200 is not a spectator sport, so the book contains almost 450exercises to challenge the reader and to deepen their understanding. The fifth edition has a number of changes: The exercises have been moved to the end of the section. The Ex-amples, Theorems, and Lemmas are now numbered in one sequenceto make it easier to find things.

3 There is a new chapter on multidimensional Brownian motion andits relationship to PDEs. To make this possible a proof of It o sformula has been added to Chapter 7. The lengthy Brownian motion chapter has been split into two, withthe second focusing on Donsker s theorem, etc. The material onthe central limit theorem for martingales and stationary sequencesdeleted from the fourth edition has been reinstated. The four sections of the random walk chapter have been times have been moved to the martingale chapter; recur-rence of random walks and the arcsine laws to the Markov chainchapter; renewal Theory has been moved to Chapter 2.

4 Some of the exercises that were simply proofs left to the reader, havebeen put into the text as lemmas. There are a few new fourth edition contains a list of the people who madecorrections to the first three editions. With apologies to those whosecontributions I lost track of, this time I need to thank: Richard Arra-tia, Benson Au, Swee Hong Chan, Conrado Costa, Nate Eldredge, SteveEvans, Jason Farnon, Christina Goldschmidt, Eduardo Horta, MartinHildebrand, Shlomo Leventhal, Jan Lieke, Kyle MacDonald, Ron Peled,Jonathan Peterson, Erfan Salavati, Byron Schmuland, Timo Seppalainen,Antonio Carlos de Azevedo Sodre, Shouda Wang, and Ruth Williams.

5 Imust confess that Christophe Leuridan pointed one out that I have notcorrected. Lemma incorrectly asserts that the distributions in itsstatement have mean 0, but their means do not exist. The conclusionremains valid since they are differentiable at 0. A sixth edition is ex-tremely unlikely, but you can email me about typos and I will post themon my web the fourth edition was being completed, Davidhad recently graduated from Ithaca College and Greg was in his lastsemester at MIT applying to graduate school in computer science.

6 Now,eight years later, Greg has graduated from Berkeley, and is an AssistantProfessor in the Computer Science department at U of Texas in works in the field of machine learning, specifically natural languageprocessing. No, I don t know what that means but it seems to pay got his degree in journalism. After an extensive job search processand some free lance work, David has settled into a steady job working fora company that produces newsletters for athletic directors and the summer of 2010, Susan and I moved to Durham.

7 Since manypeople think that the move was about the weather, I will mention thatduring our first summer it was 104 degrees (and humid!) three days in arow. Yes, it almost never snows here, but when it does, three inches ofsnow (typically mixed with ice) will shut down the whole town for fourdays. It took some time for us to adjust to the Durham/Chapel area,which has about 10 times as many people as Ithaca and is criss-crossedby freeways, but we live in a nice quiet neighborhood near the enjoys volunteering at the Sarah P.

8 Duke gardens and listening totheir talks about the plants of North Carolina and future plans for doubt there will be a sixth edition, but it is inevitable there will betypos. Email me at and I will put a list on the Durrett , January 2019 Contents1 Measure probability Spaces .. Distributions .. Random Variables .. Integration .. Properties of the Integral .. Expected Value .. Inequalities .. Integration to the Limit .. Computing Expected Values .. Product Measures, Fubini s Theorem.

9 372 Laws of Large Independence .. Sufficient Conditions for Independence .. Independence, Distribution, and Expectation .. Sums of Independent Random Variables .. Constructing Independent Random Variables .. Weak Laws of Large Numbers .. Laws .. Triangular Arrays .. Truncation .. Borel-Cantelli Lemmas .. Strong Law of Large Numbers .. Convergence of Random Series* .. Rates of Convergence .. Infinite Mean .. Renewal Theory * .. Large Deviations* .. 1053 Central Limit The De Moivre-Laplace Theorem.

10 Weak Convergence .. Examples .. Theory .. Characteristic Functions .. Definition, Inversion Formula .. Weak Convergence .. Moments and Derivatives .. Polya s Criterion* .. The Moment Problem* .. Central Limit Theorems .. Sequences .. Triangular Arrays .. Prime Divisors (Erd os-Kac)* .. Rates of Convergence (Berry-Esseen)* .. Local Limit Theorems* .. Poisson Convergence .. The Basic Limit Theorem .. Two Examples with Dependence .. Poisson Processes.