| Description |
1 online resource (xxi, 527 pages) |
| Contents |
Machine generated contents note: ch. 1 Sums of Independent Random Variables -- 1.1. Independence -- 1.1.1. Independent & sigma;-Algebras -- 1.1.2. Independent Functions -- 1.1.3. The Radomachor Functions -- Exercises for ʹ 1.1 -- 1.2. The Weak Law of Large Numbers -- 1.2.1. Orthogonal Random Variables -- 1.2.2. Independent Random Variables -- 1.2.3. Approximate Identities -- Exercises for ʹ 1.2 -- 1.3. Cramer's Theory of Large Deviations -- Exercises for ʹ 1.3 -- 1.4. The Strong Law of Large Numbers -- Exercises for ʹ 1.4 -- 1.5. Law of the Iterated Logarithm -- Exercises for ʹ 1.5 -- ch. 2 The Central Limit Theorem -- 2.1. The Basic Central Limit Theorem -- 2.1.1. Lindeberg's Theorem -- 2.1.2. The Central Limit Theorem -- Exercises for ʹ 2.1 -- 2.2. The Berry-Esseen Theorem via Stein's Method -- 2.2.1. L1-Berry-Esseen -- 2.2.2. The Classical Berry Esseen Theorem -- Exercises for ʹ 2.2 |
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2.3. Some Extensions of The Central Limit Theorem -- 2.3.1. The Fourier Transform -- 2.3.2. Multidimensional Central Limit Theorem -- 2.3.3. Higher Moments -- Exercises for ʹ 2.3 -- 2.4. An Application to Hermite Multipliers -- 2.4.1. Hermite Multipliers -- 2.4.2. Beckner's Theorem -- 2.4.3. Applications of Beckner's Theorem -- Exercises for ʹ 2.4 -- ch. 3 Infinitely Divisible Laws -- 3.1. Convergence of Measures on RN -- 3.1.1. Sequential Compactness in M1RN -- 3.1.2. Levy's Continuity Theorem -- Exercises for ʹ 3.1 -- 3.2. The Levy-Khinchine Formula -- 3.2.1. I(RN) Is the Closure of P(RN) -- 3.2.2. The Formula -- Exercises for ʹ 3.2 -- 3.3. Stable Laws -- 3.3.1. General Results -- 3.3.2. & alpha;-Stable Laws -- Exercises for ʹ 3.3 -- ch. 4 Levy Processes -- 4.1. Stochastic Processes, Some Generalities -- 4.1.1. The Space D(RN) -- 4.1.2. Jump Functions -- Exercises for ʹ 4.1 -- 4.2. Discontinuous Levy Processes -- 4.2.1. The Simple Poisson Process |
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4.2.2. Compound Poisson Processes -- 4.2.3. Poisson Jump Processes -- 4.2.4. Levy Processes with Bounded Variation -- 4.2.5. General, Non-Gaussian Levy Processes -- Exercises for ʹ 4.2 -- 4.3. Brownian Motion, the Gaussian Levy Process -- 4.3.1. Deconstructing Brownian Motion -- 4.3.2. Levy's Construction of Brownian Motion -- 4.3.3. Levy's Construction in Context -- 4.3.4. Brownian Paths Are Non-Differentiable -- 4.3.5. General Levy Processes -- Exercises for ʹ 4.3 -- ch. 5 Conditioning and Martingales -- 5.1. Conditioning -- 5.1.1. Kolmogorov's Definition -- 5.1.2. Some Extensions -- Exercises for ʹ 5.1 -- 5.2. Discrete Parameter Martingales -- 5.2.1. Doob's Inequality and Marcinkewitz's Theorem -- 5.2.2. Doob's Stopping Time Theorem -- 5.2.3. Martingale Convergence Theorem -- 5.2.4. Reversed Martingales and De Finetti's Theory -- 5.2.5. An Application to a Tracking Algorithm -- Exercises for ʹ 5.2 -- ch. 6 Some Extensions and Applications of Martingale Theory |
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6.1. Some Extensions -- 6.1.1. Martingale Theory for a & sigma;-Finite Measure Space -- 6.1.2. Banach Space -- Valued Martingales -- Exercises for ʹ 6.1 -- 6.2. Elements of Ergodic Theory -- 6.2.1. The Maximal Ergodic Lemma -- 6.2.2. Birkhoff's Ergodic Theorem -- 6.2.3. Stationary Sequences -- 6.2.4. Continuous Parameter Ergodic Theory -- Exercises for ʹ 6.2 -- 6.3. Burkholder's Inequality -- 6.3.1. Burkholder's Comparison Theorem -- 6.3.2. Burkholder's Inequality -- Exercises for ʹ 6.3 -- ch. 7 Continuous Parameter Martingales -- 7.1. Continuous Parameter Martingales -- 7.1.1. Progressively Measurable Functions -- 7.1.2. Martingales: Definition and Examples -- 7.1.3. Basic Results -- 7.1.4. Stopping Times and Stopping Theorems -- 7.1.5. An Integration by Parts Formula -- Exercises for ʹ 7.1 -- 7.2. Brownian Motion and Martingales -- 7.2.1. Levy's Characterization of Brownian Motion -- 7.2.2. Doob-Meyer Decomposition, an Easy Case -- 7.2.3. Burkholder's Inequality Again -- Exercises for ʹ 7.2 |
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7.3. The Reflection Principle Revisited -- 7.3.1. Reflecting Symmetric Levy Processes -- 7.3.2. Reflected Brownian Motion -- Exercises for ʹ 7.3 -- ch. 8 Gaussian Measures on a Banach Space -- 8.1. The Classical Wiener Space -- 8.1.1. Classical Wiener Measure -- 8.1.2. The Classical Cameron -- Martin Space -- Exercises for ʹ 8.1 -- 8.2. A Structure Theorem for Gaussian Measures -- 8.2.1. Fernique's Theorem -- 8.2.2. The Basic Structure Theorem -- 8.2.3. The Cameron -- Marin Space -- Exercises for ʹ 8.2 -- 8.3. From Hilbert to Abstract Wiener Space -- 8.3.1. An Isomorphism Theorem -- 8.3.2. Wiener Series -- 8.3.3. Orthogonal Projections -- 8.3.4. Pinned Brownian Motion -- 8.3.5. Orthogonal Invariance -- Exercises for ʹ 8.3 -- 8.4. A Large Deviations Result and Strassen's Theorem -- 8.4.1. Large Deviations for Abstract Wiener Space -- 8.4.2. Strassen's Law of the Iterated Logarithm -- Exercises for ʹ 8.4 -- 8.5. Euclidean Free Fields -- 8.5.1. The Ornstein -- Uhlenbeck Process |
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8.5.2. Ornstein -- Uhlenbeck as an Abstract Wiener Space -- 8.5.3. Higher Dimensional Free Fields -- Exercises for ʹ 8.5 -- 8.6. Brownian Motion on a Banach Space -- 8.6.1. Abstract Wiener Formulation -- 8.6.2. Brownian Formulation -- 8.6.3. Strassen's Theorem Revisited -- Exercises for ʹ 8.6 -- ch. 9 Convergence of Measures on a Polish Space -- 9.1. Prohorov -- Varadarajan Theory -- 9.1.1. Some Background -- 9.1.2. The Weak Topology -- 9.1.3. The Levy Metric and Completeness of M1(E) -- Exercises for ʹ 9.1 -- 9.2. Regular Conditional Probability Distributions -- 9.2.1. Fibering a Measure -- 9.2.2. Representing Levy Measures via the Ito Map -- Exercises for ʹ 9.2 -- 9.3. Donsker's Invariance Principle -- 9.3.1. Donsker's Theorem -- 9.3.2. Rayleigh's Random Flights Model -- Exercise for ʹ 9.3 -- ch. 10 Wiener Measure and Partial Differential Equations -- 10.1. Martingales and Partial Differential Equations -- 10.1.1. Localizing and Extending Martingale Representations |
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10.1.2. Minimum Principles -- 10.1.3. The Hermite Heat Equation -- 10.1.4. The Arcsine Law -- 10.1.5. Recurrence and Transience of Brownian Motion -- Exercises for ʹ 10.1 -- 10.2. The Markov Property and Potential Theory -- 10.2.1. The Markov Property for Wiener Measure -- 10.2.2. Recurrence in One and Two Dimensions -- 10.2.3. The Dirichlet Problem -- Exercises for ʹ 10.2 -- 10.3. Other Heat Kernels -- 10.3.1. A General Construction -- 10.3.2. The Dirichlet Heat Kernel -- 10.3.3. Feynman -- Kac Heat Kernels -- 10.3.4. Ground States and Associated Measures on Pathspace -- 10.3.5. Producing Ground States -- Exercises for ʹ 10.3 -- ch. 11 Some Classical Potential Theory -- 11.1. Uniqueness Refined -- 11.1.1. The Dirichlet Heat Kernel Again -- 11.1.2. Exiting Through & part;regG -- 11.1.3. Applications to Questions of Uniqueness -- 11.1.4. Harmonic Measure -- Exercises for ʹ 11.1 -- 11.2. The Poisson Problem and Green Functions -- 11.2.1. Green Functions when N & ge; 3 |
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11.2.2. Green Functions when N & psi; {1,2} -- Exercises for ʹ 11.2 -- 11.3. Excessive Functions, Potentials, and Riesz Decompositions -- 11.3.1. Excessive Functions -- 11.3.2. Potentials and Riesz Decomposition -- Exercises for ʹ 11.3 -- 11.4. Capacity -- 11.4.1. The Capacitory Potential -- 11.4.2. The Capacitory Distribution -- 11.4.3. Wiener's Test -- 11.4.4. Some Asymptotic Expressions Involving Capacity -- Exercises for ʹ 11.4 |
| Summary |
"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 to probability 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, where they 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 motion to the analysis of partial differential equations and potential theory"-- Provided by publisher |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Probabilities.
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probability.
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MATHEMATICS -- Probability & Statistics -- General.
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Probabilities
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Wahrscheinlichkeit
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Wahrscheinlichkeitstheorie
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Wahrscheinlichkeit.
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Wahrscheinlichkeitstheorie.
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| Form |
Electronic book
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| LC no. |
2010027652 |
| ISBN |
9781139011884 |
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113901188X |
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9781139011099 |
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113901109X |
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9780511974243 |
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0511974248 |
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9781139010825 |
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1139010824 |
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