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El. knyga: Measure Theory, Probability, and Stochastic Processes

  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 295
  • Išleidimo metai: 29-Oct-2022
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783031142055
Kitos knygos pagal šią temą:
Measure Theory, Probability, and Stochastic ProcessesDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 295
  • Išleidimo metai: 29-Oct-2022
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783031142055
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This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.





Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selectionof illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix.





Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the authors more advanced textbook in the same series (GTM 274).





 

Recenzijos

This book is a good choice for a course introducing both measure theory and probability. It would also be particularly good for self-study by those wishing to see how ideas from measure theory affect our understanding of probability and vice versa. There are exercises at the end of each chapter: some are relatively straightforward, while others expand on topics mentioned briefly in the text. (Thomas Polaski, Mathematical Reviews, August, 2024)





This book covers probability theory and stochastic processes at a graduate level. this book is exceptionally well written in a concise manner and is suitable for individuals with a strong background in undergraduate real analysis and undergraduate probability. (Eunghyun Lee, zbMATH 1526.60001, 2024)

Part I Measure Theory
1 Measurable Spaces
3(14)
1.1 Measurable Sets
3(3)
1.2 Positive Measures
6(4)
1.3 Measurable Functions
10(3)
1.4 Monotone Class
13(2)
1.5 Exercises
15(2)
2 Integration of Measurable Functions
17(24)
2.1 Integration of Nonnegative Functions
17(10)
2.2 Integrable Functions
27(4)
2.3 Integrals Depending on a Parameter
31(4)
2.4 Exercises
35(6)
3 Construction of Measures
41(22)
3.1 Outer Measures
41(3)
3.2 Lebesgue Measure
44(9)
3.3 Relation with Riemann Integrals
53(2)
3.4 A Subset of 1 Which Is Not Measurable
55(1)
3.5 Finite Measures on K and the Stieltjes Integral
56(2)
3.6 The Riesz-Markov-Kakutani Representation Theorem
58(1)
3.7 Exercises
59(4)
4 Lp Spaces
63(22)
4.1 Definitions and the Holder Inequality
63(4)
4.2 The Banach Space LP (E, A, μ)
67(4)
4.3 Density Theorems in Lp Spaces
71(4)
4.4 The Radon-Nikodym Theorem
75(6)
4.5 Exercises
81(4)
5 Product Measures
85(20)
5.1 Product a - Fields
85(2)
5.2 Product Measures
87(3)
5.3 The Fubini Theorems
90(4)
5.4 Applications
94(7)
5.4.1 Integration by Parts
94(1)
5.4.2 Convolution
95(4)
5.4.3 The Volume of the Unit Ball
99(2)
5.5 Exercises
101(4)
6 Signed Measures
105(16)
6.1 Definition and Total Variation
105(4)
6.2 The Jordan Decomposition
109(4)
6.3 The Duality Between Lp and Lq
113(5)
6.4 The Riesz-Markov-Kakutani Representation Theorem for Signed Measures
118(1)
6.5 Exercises
119(2)
7 Change of Variables
121(14)
7.1 The Change of Variables Formula
121(6)
7.2 The Gamma Function
127(1)
7.3 Lebesgue Measure on the Unit Sphere
128(2)
7.4 Exercises
130(5)
Part II Probability Theory
8 Foundations of Probability Theory
135(32)
8.1 General Definitions
136(15)
8.1.1 Probability Spaces
136(2)
8.1.2 Random Variables
138(2)
8.1.3 Mathematical Expectation
140(4)
8.1.4 An Example: Bertrand's Paradox
144(2)
8.1.5 Classical Laws
146(3)
8.1.6 Distribution Function of a Real Random Variable
149(1)
8.1.7 The a-Field Generated by a Random Variable
150(1)
8.2 Moments of Random Variables
151(11)
8.2.1 Moments and Variance
151(4)
8.2.2 Linear Regression
155(1)
8.2.3 Characteristic Functions
156(4)
8.2.4 Laplace Transform and Generating Functions
160(2)
8.3 Exercises
162(5)
9 Independence
167(32)
9.1 Independent Events
168(1)
9.2 Independence for σ-Fields and Random Variables
169(8)
9.3 The Borel-Cantelli Lemma
177(4)
9.4 Construction of Independent Sequences
181(1)
9.5 Sums of Independent Random Variables
182(4)
9.6 Convolution Semigroups
186(2)
9.7 The Poisson Process
188(7)
9.8 Exercises
195(4)
10 Convergence of Random Variables
199(28)
10.1 The Different Notions of Convergence
199(5)
10.2 The Strong Law of Large Numbers
204(5)
10.3 Convergence in Distribution
209(7)
10.4 Two Applications
216(7)
10.4.1 The Convergence of Empirical Measures
216(3)
10.4.2 The Central Limit Theorem
219(2)
10.4.3 The Multidimensional Central Limit Theorem
221(2)
10.5 Exercises
223(4)
11 Conditioning
227(30)
11.1 Discrete Conditioning
227(3)
11.2 The Definition of Conditional Expectation
230(8)
11.2.1 Integrable Random Variables
230(3)
11.2.2 Nonnegative Random Variables
233(4)
11.2.3 The Special Case of Square Integrable Variables
237(1)
11.3 Specific Properties of the Conditional Expectation
238(4)
11.4 Evaluation of Conditional Expectation
242(6)
11.4.1 Discrete Conditioning
242(1)
11.4.2 Random Variables with a Density
242(2)
11.4.3 Gaussian Conditioning
244(4)
11.5 Transition Probabilities and Conditional Distributions
248(3)
11.6 Exercises
251(6)
Part III Stochastic Processes
12 Theory of Martingales
257(46)
12.1 Definitions and Examples
257(6)
12.2 Stopping Times
263(3)
12.3 Almost Sure Convergence of Martingales
266(8)
12.4 Convergence in Lp When p > 1
274(6)
12.5 Uniform Integrability and Martingales
280(4)
12.6 Optional Stopping Theorems
284(6)
12.7 Backward Martingales
290(6)
12.8 Exercises
296(7)
13 Markov Chains
303(46)
13.1 Definitions and First Properties
303(5)
13.2 A Few Examples
308(3)
13.2.1 Independent Random Variables
308(1)
13.2.2 Random Walks on 1d
309(1)
13.2.3 Simple Random Walk on a Graph
309(1)
13.2.4 Galton-Watson Branching Processes
310(1)
13.3 The Canonical Markov Chain
311(6)
13.4 The Classification of States
317(9)
13.5 Invariant Measures
326(6)
13.6 Ergodic Theorems
332(6)
13.7 Martingales and Markov Chains
338(5)
13.8 Exercises
343(6)
14 Brownian Motion
349(46)
14.1 Brownian Motion as a Limit of Random Walks
349(4)
14.2 The Construction of Brownian Motion
353(6)
14.3 The Wiener Measure
359(2)
14.4 First Properties of Brownian Motion
361(4)
14.5 The Strong Markov Property
365(7)
14.6 Harmonic Functions and the Dirichlet Problem
372(11)
14.7 Harmonic Functions and Brownian Motion
383(6)
14.8 Exercises
389(6)
A A Few Facts from Functional Analysis 395(4)
Notes and Suggestions for Further Reading 399(2)
References 401(2)
Index 403
Jean-Franēois Le Gall is Professor of Mathematics at the University of Paris-Saclay in France. As one of the leading experts in probability theory, he has done extensive research on stochastic processes, including Brownian motion, random trees, random planar maps, and other related objects. His research accomplishments have been recognized with various awards, most recently the Wolf prize. He is the author of two successful textbooks on Brownian Motion, Martingales, and Stochastic Calculus (2016) in the Graduate Texts in Mathematics series and Spatial Branching Processes, Random Snakes and Partial Differential Equations (1999) in the Lectures in Mathematics, ETH Zürich series.
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