Probability：Theory and Examples.pdf概率论非常经典的书籍，包括大数

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详细说明：概率论非常经典的书籍，包括大数定律、中心极限定理、随机游动、鞅、马尔可夫链、遍历定理和布朗运动。它是一种综合性的处理方法，集中在对应用最有用的结果上。它的哲学是，学习概率的最好方法是看到它的实际行动，所以有200个例子和450个问题。Preface Some times the lights are shining on me. Other times I can barely see Lately it occurs to me what a long strange trip its been ful dead In 1989 when the first, edition of the book was completed, my sons David and greg were 3 and 1, and the cover picture showed the Dow Jones at 2650. The last twenty years have brought many changes but the song remains the same. The title of the book indicates that as we develop the theory, we will focus our attention on examples Hoping that the book would be a useful reference for people who apply probability in their work, we have tried to emphasize the results that are important for applications, and illustrated their use with roughly 200 examples. Probability is not a spectator sport. so the book contains almost 450 exercises to challenge the reader and to deepen their understanding The fourth edition has two major changes(in addition to a new publisher (i) The book has been converted from TeX to LaTeX. The systematic use of labels should eventually climinate problcms with rcfcrcnccs to other points in the text. In from the third edition to be reintroduced, and a number of new ones to be a dded 2 additiOnl, the picture environent and graphicx package has allowed for the figures lo (ii) Four sections of the old appendix have been combined with the first three section of Chapter 1 to Iake a llew first chapter on Measure theory, which should allow the Dook to be used by people who do not have this background without making the text. tedious for those who have AcknlowledgeInents. I aIn always grateful to the Illally people who sent ine comments and typos. Helping to correct the first edition were David Aldous, Ker Alcxandcr, Daren Clinc, Ted Cox, Robort Dalang, Joc Glover, David Griffcath, Phil Ken ross, Byron Schmuland, Steve Samuels, Jon Wellner, and Ruth Williams eres The third edition benefitted from input from Manel Baucells, Eric blair, Zhen Qing Chen, Ted Cox, Bradford Crain, Winston Crandall, Finn Christensen. Amir Dembo, Neil Falkner, Changyong Feng, Brighten Godfrey, Boris Granovsky, Jan Hannig, Andrew Hayen, Martin Hildebrand, Kyoungmun Jang: Anatole Joffe, Daniel Kifor, Stevc Kronc, Greg Lawlcr, T.Y. Lcc, Shlomo Levcntal, Torgny Lindvall, Arif Mardin, Carl Mueller, Robin Pemantle, Yuval Peres, Mark Pinsky, Ross Pinsky, Boris Pittel, David Pokorny, Vinayak Prabhu, Brett Presnell, Jim Propp, Yossi Schwarz fuchs, Rami Shakarchi, Lian Shen, Marc Shivers, Rich Sowers, Bob Strain, Tsachy Weissman, and llao zhan New helpers for the fourth edition include John Angus, Phillipe Harmony, Adam Cruz, Ricky Der, Justin Dyer, Piet Groeneboom, Vlad Island, Elena Kosygina, Richard Laugesen, Sungchul Lee, Shlomo Levental, Ping Li, Fredddy lopez, Piotr Milos, Davey Owen, Brett Presnell, Alex Smith, Harsha Wabgaonkar, John Walsh, Tsachy Weissman, Neil Wu, Ofer Zeitouni, Martin Zerner, Andrei Zherebtsov. I apologize to those whose names have been omitted or are new typos Family Update. David graduated from Ithaca College in May 2009 with a degree in print journalism and like many of his peers is struggling to find work. Greg has one semester to go at mit and is applying to graduate schools in computer science He says he wants to do research in"machine learning, so perhaps he can wrote a program to find and correct the typos in my books After 25 years in Ithaca, my wife Susan and I are moving to Durham next sum Iner,so I can take a position in the lllath departIment at Duke. Everyone seellIs to focus on the fact that we are trading very cold winters for hotter summers and a much longer growing season. but the real attraction is the excellent opportunities for interdisciplinary research in the Research Triangle The more things change the more they stay the same: inevitably there will be typos in the new version. My new coordinates are not yet set but I am sure google can find me Rick Durrett, January 2010 Contents 1 Measure Theory 1. 1 Probability spaces 1.2 Distributions 1.3 Random variables 12 1.4 Integration 15 1.5P ties of the integra. I 21 1.6E d valu 24 1.6.1 Inequalities 24 1. 6.2 Integration to the limit 1.6.3 Computing Expected values 2 1.7 Product Measures. Fubini's Theorem 31 2 Laws of Large Numbers 37 2.1 Independence ....37 2.1.1 Sufficient Conditions for Independence 2.1.2 Independence, Distribution, and Expect ation 41 2.1.3 Sums of Independent random Variables 2.1.4 Constructing Independent Random Variables 45 2.2 Weak laws of large nuimbers 2.2.2 Triangular Arrays 2.2.3 Truncation 2.3 Borcl-Cantclli camas 6 2578 ence of Random Series·,… 2.4 Strong dw c urge Nuinbers 71 2.5.2 Infinite Meal 73 2.6 Large Deviations* 3 Central limit Theorems 81 3. 1 The De Moivre-Laplace TheoreM 81 3.2 Weak Convergence 3.2.1 Examples 3.2.2 Theory 86 3.3 Characteristic Functions 91 3.3.1 Definition. Inversion formula. 91 3.3.2 Weak Convergence 9 3.3.3 Moments and derivatives 3.3.4 Polya,'s Criterion*k 101 CONTENTS 3.3.5 The moment problem* 103 3.1 Central Limit Theorems 106 3.4.1 i.i.d. Sequences 106 3.4.2 Triangular Arrays 110 3.4.3 Prime di Erdos-Kac) 3.4.4 Rates of Convergence(Berry-Esseen)* 118 3.5 Local limit theorems 121 3.6 Poisson Convergence 3.6.1 The basic Limit Theorem 126 3.6.2 Two Examples with Dependence 130 3.6.3 Poisson Processes 132 3. 7 Stable Law 135 3.8 Infinitely Divisible Distributions 3.9 Limit Theorems in rd 4 Randoin walk 153 4.1 Stopping times 4.2Re 162 4.3 Visits to 0. Arcsine laws 4.4 Renewal Theory* 5 Martingales 189 5.1 Conditional E 189 Examples 191 5.1.2Pro roperties 193 51.3R 197 5.2 Martingales, Almost Sure Convergence 198 5.3 Example 204 5.3.1 Bounded increments 204 5.3.2 Polva's Urn Scheme 205 5.3.3 Radon-Nikodym Derivatives 5.3.4 Branching Processes 209 5. 4 Dooh's Inequa lity, Converge P 5.4.1 Square Integrable Martingales ....216 5.5 Uniform Integrability, Convergence in LI 220 5.6 Backwards Martingales 225 5.7 Optional Stopping theorems 229 6 Markov Chains 233 6.1 Definitions 233 6.2 Example 6.3 Extensions of the Markov Property 6.4 Recurrence and Transience 245 6.5 Stationary Measures 51 6.6 Asymptotic Behavior 260 6.7 Periodicity, Tail o-field 266 6.8 General 270 6.8.1 Recurrence and Transience 6.8.2 Static 274 6.8.3 Convergence Theorem 275 6.8.4 GI/G/1 queue 75 CONTENTS T Ergodic Theorems 279 7. 1 Definitions and Examples 279 7.2 Birkhoff's Ergodic Theorem 283 7.4 A Subadditive Ergodic Theorems 7. 3 Recurrence 287 290 7.5 Applications 8 Brownian motion 301 8.1 Definition and construction 301 8.2 Markov Property, Blumenthal's 0-1Law 307 8.3 Stopping Times, Strong Markov Property 312 8.4 Path Properites 8. 4.1 Zeros of Brownian motion 316 8.4.2 Hitting times 316 8.4.3 Levy s Modulus of Continuity 319 8.5 Matinga.les 320 8.5.1 Multidimensional brownian motion 324 8.6 Donsker’ s Theoren 326 8.7 Empirical Distributions, Brownian Bridge 333 8.8 Laws of the Iterated Logarithm* 338 a Mcasurc Thcory details 343 A 1 Caratheeodory's Extension Theorem .343 A2 which sets are measurable? 348 A 3 Kolmogorov' s Extension Theorem 350 A 4 Radon-Nikodym Theorem 352 A.5 Differentiating Under the Integral 355 CONTENTS Chapter 1 Measure theory In this chapter, we will recall some definitions and results from measure theory. Our purpose here is to provide an introduction for readers who have not seen these concept before and to review that material for those who have. Harder proofs, especially those that do not contribute much to one's intuition, are hidden away in the appendix Readers with a solid background in mcasurc thcory can skip Scctions 1.4, 1.5, and 1.7, which were previously part of the appendix 1.1 Probability spaces Here and throughout the book terms being defined are set in boldface. We begin with the most basic quantity. A probability space is a triple (Q2, F, P)where Q is a set of“ outcomes,” f is a set of events,”andP:F→[0,1] is a function that assigns probabilities to events. We assume that F is a a-field(or a-algebra),i.e,a (nonempty) collection of subsets of 2 that satisfy ifA∈then4c∈,and (i)ifA;∈ F is a countable sequence of sets then U:A∈万. Here and in what follows, count able means finite or countably infinite. Since (i A (UiAn, it follows that a o-field is closed under countable intersections. We omit the last property from the definition to make it easier to check Without P,(Q2, F)is called a measurable space, i.e., it is a space on which we can put a measure. A measure is a nonnegative countably additive set function; that s, a function:→ R with (i)(A)≥p(0)=0 for all a∈,and (ii) if Ai E F is a countable sequence of disjoint sets, then A)=∑(A) If u(Q2)=l, we call u a probability measure. In this book, probability measures are usually denoted by P The next result gives some consequences of the definition of a measure that we will need later. In all cases we assume that the sets we mention are in F CHIAPTER 1. MEASURE TIIEORY Theorem 1.1.1. Let u be a measure on( Q2, F) (i) monotonicity. If A CB then u(A)1,Bn≡A=Um=1(A1m). Since the Bn are disjoint and have union A we have using (1) of the definition of measure, Bm C Am. and (i) of this theorem 1(A)=∑(Bn)∑(An) m-1 (iii) Let Bn An An-I. Then the Bn arc disjoint and havc Umm== 4, m=iBm= An (4)=∑(B ∑风B u(An) (iv)A1-An↑A1-Aso(i) impliesμ(41-An)↑p(A1=A). Since Ab b we have A(Al B)=u(A1 u(B) and it follows that u(An)L u(A) The simplest setting, which should be familiar from undergraduate probability, is Example 1.1.1. Discrete probability spaces. Let Q= a countable set, i.e., finite or countably infinite. Let F= the set of all subsets of Q2. Let ∑p(u)whcp(a)≥0and∑p(a)=1 ∈A A little thought reveals that t his is the most genera. I probability measure on this space. In many cases when Q2 is a finite set, we have p(w)=1/Q2 where 9= the number of points in g For a simple concrete example that requires this level of generality consider the astragali, dice used in ancient Egypt made from the ankle bones of sheep. This die ould comc to rest on the top sidc of the bonc for four points or on the bottom for three points. The side of the bone was slightly rounded. The die could cone on a fat and narrow piece for six points or somewhere on the rest of the side for one point. There is no reason to think that all four outcomes are equally likely so we need probabilities p1, p3, p4, and p6 to describe P To prepare for our next definition, we need Exercise11.1.(i)If丌,i∈Iareσ- fields then∩∈nJis. HIere l≠ is an arbitrary index set (i. e, possibly uncountable). (ii) Use the result in(i) to show if we are given a set S2 and a collection of subsets of a, then there is a smallest o-field containing A. We will call this the a-field generated by A and denote it by a(A)

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