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Probability Theory: An Analytic View 2nd Revised edition [Hardback]

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(Massachusetts Institute of Technology)
  • Formāts: Hardback, 550 pages, height x width x depth: 260x185x33 mm, weight: 1090 g, Worked examples or Exercises
  • Izdošanas datums: 31-Dec-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521761581
  • ISBN-13: 9780521761581
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Probability Theory: An Analytic View 2nd Revised editionPalielināt
  • Formāts: Hardback, 550 pages, height x width x depth: 260x185x33 mm, weight: 1090 g, Worked examples or Exercises
  • Izdošanas datums: 31-Dec-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521761581
  • ISBN-13: 9780521761581
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521761581.html
Keywords:
"This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction toprobability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, wherethey are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motionto the analysis of partial differential equations and potential theory"--

Recenzijas

' uniformly well written and well spiced with comments to aid the intuition, so the readership should include a wide range, both of students and of professional probabilists. We can expect it to take its place alongside the classics of probability theory.' Mathematical Reviews

Papildus informācija

A second edition of Daniel W. Stroock's classic probability theory textbook suitable for first-year graduate students with a good grasp of introductory, undergraduate probability.
Preface xiii
Table of Dependence
xxi
Chapter 1 Sums of Independent Random Variables
1(58)
1.1 Independence
1(13)
1.1.1 Independent σ-Algebras
1(3)
1.1.2 Independent Functions
4(1)
1.1.3 The Radomachor Functions
5(2)
Exercises for § 1.1
7(7)
1.2 The Weak Law of Large Numbers
14(6)
1.2.1 Orthogonal Random Variables
14(1)
1.2.2 Independent Random Variables
15(1)
1.2.3 Approximate Identities
16(4)
Exercises for § 1.2
20(2)
1.3 Cramer's Theory of Large Deviations
22(13)
Exercises for § 1.3
31(4)
1.4 The Strong Law of Large Numbers
35(14)
Exercises for § 1.4
42(7)
1.5 Law of the Iterated Logarithm
49(10)
Exercises for § 1.5
56(3)
Chapter 2 The Central Limit Theorem
59(56)
2.1 The Basic Central Limit Theorem
60(11)
2.1.1 Lindeberg's Theorem
60(2)
2.1.2 The Central Limit Theorem
62(3)
Exercises for § 2.1
65(6)
2.2 The Berry-Esseen Theorem via Stein's Method
71(11)
2.2.1 L1-Berry-Esseen
72(3)
2.2.2 The Classical Berry Esseen Theorem
75(6)
Exercises for § 2.2
81(1)
2.3 Some Extensions of The Central Limit Theorem
82(14)
2.3.1 The Fourier Transform
82(2)
2.3.2 Multidimensional Central Limit Theorem
84(3)
2.3.3 Higher Moments
87(3)
Exercises for § 2.3
90(6)
2.4 An Application to Hermite Multipliers
96(14)
2.4.1 Hermite Multipliers
96(5)
2.4.2 Beckner's Theorem
101(4)
2.4.3 Applications of Beckner's Theorem
105(5)
Exercises for § 2.4
110(5)
Chapter 3 Infinitely Divisible Laws
115(36)
3.1 Convergence of Measures on RN
5(117)
3.1.1 Sequential Compactness in M1RN
116(1)
3.1.2 Levy's Continuity Theorem
117(2)
Exercises for § 3.1
119(3)
3.2 The Levy-Khinchine Formula
122(17)
3.2.1 I(RN) Is the Closure of P(RN)
123(3)
3.2.2 The Formula
126(11)
Exercises for § 3.2
137(2)
3.3 Stable Laws
139(12)
3.3.1 General Results
139(2)
3.3.2 α-Stable Laws
141(6)
Exercises for § 3.3
147(4)
Chapter 4 Levy Processes
151(42)
4.1 Stochastic Processes, Some Generalities
152(8)
4.1.1 The Space D(RN)
153(3)
4.1.2 Jump Functions
156(3)
Exercises for § 4.1
159(1)
4.2 Discontinuous Levy Processes
160(17)
4.2.1 The Simple Poisson Process
161(2)
4.2.2 Compound Poisson Processes
163(5)
4.2.3 Poisson Jump Processes
168(2)
4.2.4 Levy Processes with Bounded Variation
170(1)
4.2.5 General, Non-Gaussian Levy Processes
171(3)
Exercises for § 4.2
174(3)
4.3 Brownian Motion, the Gaussian Levy Process
177(16)
4.3.1 Deconstructing Brownian Motion
178(2)
4.3.2 Levy's Construction of Brownian Motion
180(2)
4.3.3 Levy's Construction in Context
182(1)
4.3.4 Brownian Paths Are Non-Differentiable
183(2)
4.3.5 General Levy Processes
185(2)
Exercises for § 4.3
187(6)
Chapter 5 Conditioning and Martingales
193(40)
5.1 Conditioning
193(12)
5.1.1 Kolmogorov's Definition
194(4)
5.1.2 Some Extensions
198(4)
Exercises for § 5.1
202(3)
5.2 Discrete Parameter Martingales
205(28)
5.2.1 Doob's Inequality and Marcinkewitz's Theorem
206(6)
5.2.2 Doob's Stopping Time Theorem
212(2)
5.2.3 Martingale Convergence Theorem
214(3)
5.2.4 Reversed Martingales and De Finetti's Theory
217(4)
5.2.5 An Application to a Tracking Algorithm
221(5)
Exercises for § 5.2
226(7)
Chapter 6 Some Extensions and Applications of Martingale Theory
233(33)
6.1 Some Extensions
233(11)
6.1.1 Martingale Theory for a σ-Finite Measure Space
233(6)
6.1.2 Banach Space--Valued Martingales
239(1)
Exercises for § 6.1
240(4)
6.2 Elements of Ergodic Theory
244(13)
6.2.1 The Maximal Ergodic Lemma
245(3)
6.2.2 Birkhoff's Ergodic Theorem
248(3)
6.2.3 Stationary Sequences
251(2)
6.2.4 Continuous Parameter Ergodic Theory
253(3)
Exercises for § 6.2
256(1)
6.3 Burkholder's Inequality
257(9)
6.3.1 Burkholder's Comparison Theorem
257(5)
6.3.2 Burkholder's Inequality
262(1)
Exercises for § 6.3
263(3)
Chapter 7 Continuous Parameter Martingales
266(33)
7.1 Continuous Parameter Martingales
266(16)
7.1.1 Progressively Measurable Functions
266(1)
7.1.2 Martingales: Definition and Examples
267(3)
7.1.3 Basic Results
270(2)
7.1.4 Stopping Times and Stopping Theorems
272(4)
7.1.5 An Integration by Parts Formula
276(4)
Exercises for § 7.1
280(2)
7.2 Brownian Motion and Martingales
282(10)
7.2.1 Levy's Characterization of Brownian Motion
282(2)
7.2.2 Doob-Meyer Decomposition, an Easy Case
284(5)
7.2.3 Burkholder's Inequality Again
289(1)
Exercises for § 7.2
290(2)
7.3 The Reflection Principle Revisited
292(7)
7.3.1 Reflecting Symmetric Levy Processes
292(2)
7.3.2 Reflected Brownian Motion
294(4)
Exercises for § 7.3
298(1)
Chapter 8 Gaussian Measures on a Banach Space
299(68)
8.1 The Classical Wiener Space
299(7)
8.1.1 Classical Wiener Measure
299(4)
8.1.2 The Classical Cameron--Martin Space
303(3)
Exercises for § 8.1
306(1)
8.2 A Structure Theorem for Gaussian Measures
306(11)
8.2.1 Fernique's Theorem
306(1)
8.2.2 The Basic Structure Theorem
307(3)
8.2.3 The Cameron--Marin Space
310(3)
Exercises for § 8.2
313(4)
8.3 From Hilbert to Abstract Wiener Space
317(20)
8.3.1 An Isomorphism Theorem
317(1)
8.3.2 Wiener Series
318(4)
8.3.3 Orthogonal Projections
322(4)
8.3.4 Pinned Brownian Motion
326(2)
8.3.5 Orthogonal Invariance
328(2)
Exercises for § 8.3
330(7)
8.4 A Large Deviations Result and Strassen's Theorem
337(6)
8.4.1 Large Deviations for Abstract Wiener Space
337(3)
8.4.2 Strassen's Law of the Iterated Logarithm
340(2)
Exercises for § 8.4
342(1)
8.5 Euclidean Free Fields
343(15)
8.5.1 The Ornstein--Uhlenbeck Process
344(2)
8.5.2 Ornstein--Uhlenbeck as an Abstract Wiener Space
346(3)
8.5.3 Higher Dimensional Free Fields
349(6)
Exercises for § 8.5
355(3)
8.6 Brownian Motion on a Banach Space
358(9)
8.6.1 Abstract Wiener Formulation
358(3)
8.6.2 Brownian Formulation
361(2)
8.6.3 Strassen's Theorem Revisited
363(2)
Exercises for § 8.6
365(2)
Chapter 9 Convergence of Measures on a Polish Space
367(33)
9.1 Prohorov--Varadarajan Theory
367(19)
9.1.1 Some Background
367(3)
9.1.2 The Weak Topology
370(7)
9.1.3 The Levy Metric and Completeness of M1(E)
377(4)
Exercises for § 9.1
381(5)
9.2 Regular Conditional Probability Distributions
386(6)
9.2.1 Fibering a Measure
388(2)
9.2.2 Representing Levy Measures via the Ito Map
390(2)
Exercises for § 9.2
392(1)
9.3 Donsker's Invariance Principle
392(8)
9.3.1 Donsker's Theorem
393(3)
9.3.2 Rayleigh's Random Flights Model
396(3)
Exercise for § 9.3
399(1)
Chapter 10 Wiener Measure and Partial Differential Equations
400(56)
10.1 Martingales and Partial Differential Equations
400(16)
10.1.1 Localizing and Extending Martingale Representations
401(3)
10.1.2 Minimum Principles
404(1)
10.1.3 The Hermite Heat Equation
405(2)
10.1.4 The Arcsine Law
407(4)
10.1.5 Recurrence and Transience of Brownian Motion
411(4)
Exercises for § 10.1
415(1)
10.2 The Markov Property and Potential Theory
416(13)
10.2.1 The Markov Property for Wiener Measure
416(1)
10.2.2 Recurrence in One and Two Dimensions
417(1)
10.2.3 The Dirichlet Problem
418(8)
Exercises for § 10.2
426(3)
10.3 Other Heat Kernels
429(27)
10.3.1 A General Construction
429(2)
10.3.2 The Dirichlet Heat Kernel
431(5)
10.3.3 Feynman--Kac Heat Kernels
436(3)
10.3.4 Ground States and Associated Measures on Pathspace
439(6)
10.3.5 Producing Ground States
445(4)
Exercises for § 10.3
449(7)
Chapter 11 Some Classical Potential Theory
456(61)
11.1 Uniqueness Refined
456(19)
11.1.1 The Dirichlet Heat Kernel Again
456(3)
11.1.2 Exiting Through ∂regG
459(4)
11.1.3 Applications to Questions of Uniqueness
463(5)
11.1.4 Harmonic Measure
468(4)
Exercises for § 11.1
472(3)
11.2 The Poisson Problem and Green Functions
475(12)
11.2.1 Green Functions when N ≥ 3
476(1)
11.2.2 Green Functions when N ψ {1,2}
477(9)
Exercises for § 11.2
486(1)
11.3 Excessive Functions, Potentials, and Riesz Decompositions
487(10)
11.3.1 Excessive Functions
488(1)
11.3.2 Potentials and Riesz Decomposition
489(7)
Exercises for § 11.3
496(1)
11.4 Capacity
497(20)
11.4.1 The Capacitory Potential
497(3)
11.4.2 The Capacitory Distribution
500(4)
11.4.3 Wiener's Test
504(3)
11.4.4 Some Asymptotic Expressions Involving Capacity
507(7)
Exercises for § 11.4
514(3)
Notation 517(4)
Index 521
Dr Daniel W. Stroock is the Simons Professor of Mathematics Emeritus at the Massachusetts Institute of Technology. He has published numerous articles and is the author of six books, most recently Partial Differential Equations for Probabilists (2008).