Probability Theory and Examples.pdf一本很好的讲概率的书，有很多例

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详细说明：一本很好的讲概率的书，有很多例子，讲了测度理论和鞅论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 2.5.sEnce of Random Series*x 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 array 3.4.3 Prime Divisors(Erdos-Kac) 114 3.4.4 Rates of Convergence(Berry-Esseen)* 118 3.5 Local limit theorems 121 3.6 Poisson Convergence 126 3.6.1 The Basic limit Theorem 126 39 Limit Theorems in pa" butions*·,·,、· 3.6.2 Two Examples with Dependence 130 3.6.3 Poisson processes 3.7 Stable Laws* p 132 135 3.8 Infinitely Divisible Dist 144 147 4 Random walks 153 4.1 Stopping Times 番鲁鲁 4.2 Recurrence 4.3 Visits to 0. Arcsine Laws 162 172 4.4 Renewal Theory 177 rting 189 5.1 Conditional Expectation 189 5.1.1E 191 5.1.2P 193 5.1.3 Regular Conditional Probabilities* 197 5.2 Martingales, Almost Sure Convergence 198 5.3 Examples 204 5.3.1 Bounded increments .204 5.3.2 Polva's Urn Scheme 205 5.3.3 Radon-Nikodym Derivatives 54 Doob's Inequality, Convergence in··" 5.3.4 Branching p 209 212 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 29 6 Markov chains 233 6. 1 Definitions ..233 6.2E 236 6.3 Extensions of the Markov Property 6.4 Recurrence and transien 6.5 Stationary Measures 252 6.6A ymp totic b 261 6.7 Periodicity, Tail o-field 266 6. 8 General State Space* 270 6.8.1 Recurrence and Transience 6.8.2 Stat Meas 274 6.8.3 Convergence Theorem 275 6.8.4 GI/G/1 qu 76 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 8.6Ito’ s formula* 32 8.7 Donsker's Theorem 333 8.8 CLT's for Martingales* 8.9 Empirical Distributions, Brownian Bridge 340 346 8.10 Weak convergence* 351 8.10.1Thes 35 8. 10.2 The Space D 353 8.11 Laws of the Iterated Logarithm* 355 A Measure Theory Details 359 A1 Carathe codory's Extension Thcorcm ..359 A 2 Which Sets Are measurable? 364 A3 Kolmogorov's Extension Theorem 366 A 4 Radon-Nikodym Theorem 番番 368 A. 5 Differentiating under the Integral 371 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 Q2 that satisfy A∈ f then a∈J,and (ii) if A: E F is a countable sequence of sets then U Ai EF Here and in what follows, count able means finite or countably infinite. Since ni A; )c, 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,(Q, 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 is, a function u: F-R with (i)(A)≥p(0)=0 for all a∈,and (ii) if Ai E F is a countable sequence of disjoint sets, then 4)=∑(A) If u(Q2)=1, 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(, F) (i) monotonicity. If A C b then u(a)u(B) (ii) subadditivity. If A C Umm_1 Am then u(A)<>m_i u(am) (iii) continuity froin below, I Ai 1 A(i.e, A1 C A2 C.. and UiAi= A) cheT 1(A)-p(A) (iv) continuity from above. If Ai A(i.e, A1 3 A23..and ni Ai= A), with (A1) Ai)lu(a Proof. (i)Let B-A=Bn a be the difference of the two sets. Using to denote disjoint union, B=A+(B-A)so (B)=(4)+1(B-4)≥(A) (ii)Let An= An n A. B1= A1 and for n>1, Bn= An -=Am. Since the Bn are disjoint and have union A we have using (ii) of the definition of measure, Br C Am and (i of this theorem ∑H(Bn)≤∑A(A m-1 (iii)Lct Bn= 4n- An-1. Thon thc Bn arc disjoint and havc Um_Bm =4, B 1(4)=∑1(Bm)=im∑(Bm)=1im(A (iv)A1-An t ai-a so(iii) implies u(A1-An)t u(A1-A). Since A1 > B we have u(A1-B)=u(A1)-u(b) and it follows that u(An)Iu(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) 1.1. PROBABILITY SPACES Let r be the set of vectors( 1,..d) of real numbers and r be the Borel sets, the smallest a-field containing the open sets. When d= l we drop the superscript Example 1.1.2. Measures on the real line Measures on(R, R) are defined by giving probability a stieltjes measure function with the following properties (i F is nondecreasing (ii) F is right, continous, i.e. limgla F(y)= F(r) Theorem 1.1.2. Associated with each Stieltjes measure function F there is a unique measure u on(R, R) with u((a, b)= F(b-F(a) u((a, b)= F(b)-F(a (1.1 When F(a)=m the resulting measure is called Lebesgue measure The proof of Theorem 1.1.2 is a long and winding road, so we will content ourselves to describe the main ideas involved in this section and to hide the remaining details in the appendix in Section A 1. The choice of "closed on the right?"in(a, b is dictated by the fact that if bn b then we have (a, bn]=(a, b The next definition will explain the choice of"open on the left A collection S of sets is said to be a semialgebra if (i) it is closed under inter ction,ie,S,T∈ S implies s∩T∈S,ad(i)ifs∈ S then s is a finite disj union of sets in S. An important example of a semialgebra is Example 1.1.3. Sa= the empty set plus all sets of the forIn (a1,b1]×…x(a,bd l now, let Fk=Bin..n Bi-in Bk and note ∪B;=F1+…+F A-A∩(UB1)-(A∩F1)+…+(A∩ so using(a),(b)with n= 1, and(a)again (4)=∑(A∩F)s∑(F)=n(B

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