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E-raamat: Probability: Theory and Examples

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(Duke University, North Carolina)
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Probability: Theory and ExamplesSuurem pilt
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This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for applications, this comprehensive treatment is a rigorous graduate text and reference. Operating under the philosophy that the best way to learn probability is to see it in action, the book contains extended examples that apply the theory to concrete applications. This fifth edition contains a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic that is finding new applications. Setting the foundation for this expansion, Chapter 7 now features a proof of Itō's formula. Key exercises that previously were simply proofs left to the reader have been directly inserted into the text as lemmas. The new edition re-instates discussion about the central limit theorem for martingales and stationary sequences.

Arvustused

'Probability: Theory and Examples 5th Edition still holds true to its original goal that as the theory is developed, the focus of attention will be on examples with hundreds of examples provided and hundreds of example problems given as exercises for the reader.' Brent Kelderman, MAA Reviews

Muu info

A well-written and lively introduction to measure theoretic probability for graduate students and researchers.
Preface xi
1 Measure Theory
1(36)
1.1 Probability Spaces
1(7)
1.2 Distributions
8(5)
1.3 Random Variables
13(2)
1.4 Integration
15(6)
1.5 Properties of the Integral
21(4)
1.6 Expected Value
25(8)
1.6.1 Inequalities
25(1)
1.6.2 Integration to the Limit
26(2)
1.6.3 Computing Expected Values
28(5)
1.7 Product Measures, Fubini's Theorem
33(4)
2 Laws of Large Numbers
37(61)
2.1 Independence
37(11)
2.1.1 Sufficient Conditions for Independence
39(2)
2.1.2 Independence, Distribution, and Expectation
41(2)
2.1.3 Sums of Independent Random Variables
43(2)
2.1.4 Constructing Independent Random Variables
45(3)
2.2 Weak Laws of Large Numbers
48(10)
2.2.1 L2 Weak Laws
48(3)
2.2.2 Triangular Arrays
51(2)
2.2.3 Truncation
53(5)
2.3 Borel-Cantelli Lemmas
58(7)
2.4 Strong Law of Large Numbers
65(4)
2.5 Convergence of Random Series*
69(9)
2.5.1 Rates of Convergence
75(1)
2.5.2 Infinite Mean
76(2)
2.6 Renewal Theory*
78(12)
2.7 Large Deviations*
90(8)
3 Central Limit Theorems
98(80)
3.1 The De Moivre-Laplace Theorem
98(2)
3.2 Weak Convergence
100(8)
3.2.1 Examples
100(2)
3.2.2 Theory
102(6)
3.3 Characteristic Functions
108(17)
3.3.1 Definition, Inversion Formula
108(6)
3.3.2 Weak Convergence
114(2)
3.3.3 Moments and Derivatives
116(3)
3.3.4 Polya's Criterion*
119(2)
3.3.5 The Moment Problem*
121(4)
3.4 Central Limit Theorems
125(15)
3.4.1 i.i.d. Sequences
125(3)
3.4.2 Triangular Arrays
128(4)
3.4.3 Prime Divisors (Erdos-Kac)*
132(4)
3.4.4 Rates of Convergence (Berry-Esseen)*
136(4)
3.5 Local Limit Theorems*
140(5)
3.6 Poisson Convergence
145(6)
3.6.1 The Basic Limit Theorem
145(4)
3.6.2 Two Examples with Dependence
149(2)
3.7 Poisson Processes
151(8)
3.7.1 Compound Poisson Processes
154(1)
3.7.2 Thinning
155(2)
3.7.3 Conditioning
157(2)
3.8 Stable Laws*
159(9)
3.9 Infinitely Divisible Distributions*
168(3)
3.10 Limit Theorems in Rd
171(7)
4 Martingales
178(54)
4.1 Conditional Expectation
178(10)
4.1.1 Examples
180(2)
4.1.2 Properties
182(3)
4.1.3 Regular Conditional Probabilities*
185(3)
4.2 Martingales, Almost Sure Convergence
188(6)
4.3 Examples
194(9)
4.3.1 Bounded Increments
194(2)
4.3.2 Polya's Urn Scheme
196(1)
4.3.3 Radon-Nikodym Derivatives
197(3)
4.3.4 Branching Processes
200(3)
4.4 Doob's Inequality, Convergence in Lp, p > 1
203(5)
4.5 Square Integrable Martingales*
208(3)
4.6 Uniform Integrability, Convergence in L1
211(5)
4.7 Backwards Martingales
216(5)
4.8 Optional Stopping Theorems
221(6)
4.8.1 Applications to Random Walks
223(4)
4.9 Combinatorics of Simple Random Walk*
227(5)
5 Markov Chains
232(54)
5.1 Examples
232(3)
5.2 Construction, Markov Properties
235(8)
5.3 Recurrence and Transience
243(5)
5.4 Recurrence of Random Walks Stararred Section
248(11)
5.5 Stationary Measures
259(9)
5.6 Asymptotic Behavior
268(6)
5.7 Periodicity, Tail σ-Field*
274(4)
5.8 General State Space*
278(8)
5.8.1 Recurrence and Transience
281(1)
5.8.2 Stationary Measures
281(1)
5.8.3 Convergence Theorem
282(1)
5.8.4 GI/G/1 Queue
283(3)
6 Ergodic Theorems
286(19)
6.1 Definitions and Examples
286(3)
6.2 Birkhoff's Ergodic Theorem
289(4)
6.3 Recurrence
293(3)
6.4 A Subadditive Ergodic Theorem
296(4)
6.5 Applications
300(5)
7 Brownian Motion
305(31)
7.1 Definition and Construction
305(6)
7.2 Markov Property, Blumenthal's 0-1 Law
311(5)
7.3 Stopping Times, Strong Markov Property
316(4)
7.4 Path Properties
320(5)
7.4.1 Zeros of Brownian Motion
320(1)
7.4.2 Hitting Times
321(4)
7.5 Martingales
325(3)
7.6 Ito's Formula*
328(8)
8 Applications to Random Walk
336(28)
8.1 Donsker's Theorem
336(6)
8.2 CLTs for Martingales
342(5)
8.3 CLTs for Stationary Sequences
347(7)
8.3.1 Mixing Properties
351(3)
8.4 Empirical Distributions, Brownian Bridge
354(6)
8.5 Laws of the Iterated Logarithm
360(4)
9 Multidimensional Brownian Motion
364(30)
9.1 Martingales
364(2)
9.2 Heat Equation
366(2)
9.3 Inhomogeneous Heat Equation
368(2)
9.4 Feynman-Kac Formula
370(3)
9.5 Dirichlet Problem
373(6)
9.5.1 Exit Distributions
377(2)
9.6 Green's Functions and Potential Kernels
379(3)
9.7 Poisson's Equation
382(5)
9.7.1 Occupation Times
385(2)
9.8 Schrodinger Equation
387(7)
Appendix A Measure Theory Details
394(16)
A.1 Caratheodory's Extension Theorem
394(5)
A.2 Which Sets Are Measurable?
399(3)
A.3 Kolmogorov's Extension Theorem
402(1)
A.4 Radon-Nikodym Theorem
403(4)
A.5 Differentiating under the Integral
407(3)
References 410(5)
Index 415
Rick Durrett is a James B. Duke professor in the mathematics department of Duke University, North Carolina. He received his Ph.D. in Operations Research from Stanford University in 1976. After nine years at University of California, Los Angeles and twenty-five at Cornell University, he moved to Duke University in 2010. He is the author of 8 books and more than 220 journal articles on a wide variety of topics, and has supervised more than 45 Ph.D. students. He is a member of National Academy of Science, American Academy of Arts and Sciences, and a fellow of the Institute of Mathematical Statistics, and of the American Mathematical Society.
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