Probability theory :: an analytic view /
"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...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge ; New York :
Cambridge University Press,
©2011.
|
| Ausgabe: | 2nd ed. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | "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"-- |
| Beschreibung: | 1 online resource (xxi, 527 pages) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9781139011884 113901188X 9781139011099 113901109X 9780511974243 0511974248 9781139010825 1139010824 |
Internformat
MARC
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| 049 | |a MAIN | ||
| 100 | 1 | |a Stroock, Daniel W. | |
| 245 | 1 | 0 | |a Probability theory : |b an analytic view / |c Daniel W. Stroock. |
| 250 | |a 2nd ed. | ||
| 260 | |a Cambridge ; |a New York : |b Cambridge University Press, |c ©2011. | ||
| 300 | |a 1 online resource (xxi, 527 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 520 | |a "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"-- |c Provided by publisher. | ||
| 504 | |a Includes bibliographical references and index. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | 0 | |g Machine generated contents note: |g ch. 1 |t Sums of Independent Random Variables -- |g 1.1. |t Independence -- |g 1.1.1. |t Independent & sigma;-Algebras -- |g 1.1.2. |t Independent Functions -- |g 1.1.3. |t The Radomachor Functions -- |t Exercises for ʹ 1.1 -- |g 1.2. |t The Weak Law of Large Numbers -- |g 1.2.1. |t Orthogonal Random Variables -- |g 1.2.2. |t Independent Random Variables -- |g 1.2.3. |t Approximate Identities -- |t Exercises for ʹ 1.2 -- |g 1.3. |t Cramer's Theory of Large Deviations -- |t Exercises for ʹ 1.3 -- |g 1.4. |t The Strong Law of Large Numbers -- |t Exercises for ʹ 1.4 -- |g 1.5. |t Law of the Iterated Logarithm -- |t Exercises for ʹ 1.5 -- |g ch. 2 |t The Central Limit Theorem -- |g 2.1. |t The Basic Central Limit Theorem -- |g 2.1.1. |t Lindeberg's Theorem -- |g 2.1.2. |t The Central Limit Theorem -- |t Exercises for ʹ 2.1 -- |g 2.2. |t The Berry-Esseen Theorem via Stein's Method -- |g 2.2.1. |t L1-Berry-Esseen -- |g 2.2.2. |t The Classical Berry Esseen Theorem -- |t Exercises for ʹ 2.2. |
| 505 | 0 | 0 | |g 2.3. |t Some Extensions of The Central Limit Theorem -- |g 2.3.1. |t The Fourier Transform -- |g 2.3.2. |t Multidimensional Central Limit Theorem -- |g 2.3.3. |t Higher Moments -- |t Exercises for ʹ 2.3 -- |g 2.4. |t An Application to Hermite Multipliers -- |g 2.4.1. |t Hermite Multipliers -- |g 2.4.2. |t Beckner's Theorem -- |g 2.4.3. |t Applications of Beckner's Theorem -- |t Exercises for ʹ 2.4 -- |g ch. 3 |t Infinitely Divisible Laws -- |g 3.1. |t Convergence of Measures on RN -- |g 3.1.1. |t Sequential Compactness in M1RN -- |g 3.1.2. |t Levy's Continuity Theorem -- |t Exercises for ʹ 3.1 -- |g 3.2. |t The Levy-Khinchine Formula -- |g 3.2.1. |t I(RN) Is the Closure of P(RN) -- |g 3.2.2. |t The Formula -- |t Exercises for ʹ 3.2 -- |g 3.3. |t Stable Laws -- |g 3.3.1. |t General Results -- |g 3.3.2. |t & alpha;-Stable Laws -- |t Exercises for ʹ 3.3 -- |g ch. 4 |t Levy Processes -- |g 4.1. |t Stochastic Processes, Some Generalities -- |g 4.1.1. |t The Space D(RN) -- |g 4.1.2. |t Jump Functions -- |t Exercises for ʹ 4.1 -- |g 4.2. |t Discontinuous Levy Processes -- |g 4.2.1. |t The Simple Poisson Process. |
| 505 | 0 | 0 | |g 4.2.2. |t Compound Poisson Processes -- |g 4.2.3. |t Poisson Jump Processes -- |g 4.2.4. |t Levy Processes with Bounded Variation -- |g 4.2.5. |t General, Non-Gaussian Levy Processes -- |t Exercises for ʹ 4.2 -- |g 4.3. |t Brownian Motion, the Gaussian Levy Process -- |g 4.3.1. |t Deconstructing Brownian Motion -- |g 4.3.2. |t Levy's Construction of Brownian Motion -- |g 4.3.3. |t Levy's Construction in Context -- |g 4.3.4. |t Brownian Paths Are Non-Differentiable -- |g 4.3.5. |t General Levy Processes -- |t Exercises for ʹ 4.3 -- |g ch. 5 |t Conditioning and Martingales -- |g 5.1. |t Conditioning -- |g 5.1.1. |t Kolmogorov's Definition -- |g 5.1.2. |t Some Extensions -- |t Exercises for ʹ 5.1 -- |g 5.2. |t Discrete Parameter Martingales -- |g 5.2.1. |t Doob's Inequality and Marcinkewitz's Theorem -- |g 5.2.2. |t Doob's Stopping Time Theorem -- |g 5.2.3. |t Martingale Convergence Theorem -- |g 5.2.4. |t Reversed Martingales and De Finetti's Theory -- |g 5.2.5. |t An Application to a Tracking Algorithm -- |t Exercises for ʹ 5.2 -- |g ch. 6 |t Some Extensions and Applications of Martingale Theory. |
| 505 | 0 | 0 | |g 6.1. |t Some Extensions -- |g 6.1.1. |t Martingale Theory for a & sigma;-Finite Measure Space -- |g 6.1.2. |t Banach Space -- Valued Martingales -- |t Exercises for ʹ 6.1 -- |g 6.2. |t Elements of Ergodic Theory -- |g 6.2.1. |t The Maximal Ergodic Lemma -- |g 6.2.2. |t Birkhoff's Ergodic Theorem -- |g 6.2.3. |t Stationary Sequences -- |g 6.2.4. |t Continuous Parameter Ergodic Theory -- |t Exercises for ʹ 6.2 -- |g 6.3. |t Burkholder's Inequality -- |g 6.3.1. |t Burkholder's Comparison Theorem -- |g 6.3.2. |t Burkholder's Inequality -- |t Exercises for ʹ 6.3 -- |g ch. 7 |t Continuous Parameter Martingales -- |g 7.1. |t Continuous Parameter Martingales -- |g 7.1.1. |t Progressively Measurable Functions -- |g 7.1.2. |t Martingales: Definition and Examples -- |g 7.1.3. |t Basic Results -- |g 7.1.4. |t Stopping Times and Stopping Theorems -- |g 7.1.5. |t An Integration by Parts Formula -- |t Exercises for ʹ 7.1 -- |g 7.2. |t Brownian Motion and Martingales -- |g 7.2.1. |t Levy's Characterization of Brownian Motion -- |g 7.2.2. |t Doob-Meyer Decomposition, an Easy Case -- |g 7.2.3. |t Burkholder's Inequality Again -- |t Exercises for ʹ 7.2. |
| 505 | 0 | 0 | |g 7.3. |t The Reflection Principle Revisited -- |g 7.3.1. |t Reflecting Symmetric Levy Processes -- |g 7.3.2. |t Reflected Brownian Motion -- |t Exercises for ʹ 7.3 -- |g ch. 8 |t Gaussian Measures on a Banach Space -- |g 8.1. |t The Classical Wiener Space -- |g 8.1.1. |t Classical Wiener Measure -- |g 8.1.2. |t The Classical Cameron -- Martin Space -- |t Exercises for ʹ 8.1 -- |g 8.2. |t A Structure Theorem for Gaussian Measures -- |g 8.2.1. |t Fernique's Theorem -- |g 8.2.2. |t The Basic Structure Theorem -- |g 8.2.3. |t The Cameron -- Marin Space -- |t Exercises for ʹ 8.2 -- |g 8.3. |t From Hilbert to Abstract Wiener Space -- |g 8.3.1. |t An Isomorphism Theorem -- |g 8.3.2. |t Wiener Series -- |g 8.3.3. |t Orthogonal Projections -- |g 8.3.4. |t Pinned Brownian Motion -- |g 8.3.5. |t Orthogonal Invariance -- |t Exercises for ʹ 8.3 -- |g 8.4. |t A Large Deviations Result and Strassen's Theorem -- |g 8.4.1. |t Large Deviations for Abstract Wiener Space -- |g 8.4.2. |t Strassen's Law of the Iterated Logarithm -- |t Exercises for ʹ 8.4 -- |g 8.5. |t Euclidean Free Fields -- |g 8.5.1. |t The Ornstein -- Uhlenbeck Process. |
| 505 | 0 | 0 | |g 8.5.2. |t Ornstein -- Uhlenbeck as an Abstract Wiener Space -- |g 8.5.3. |t Higher Dimensional Free Fields -- |t Exercises for ʹ 8.5 -- |g 8.6. |t Brownian Motion on a Banach Space -- |g 8.6.1. |t Abstract Wiener Formulation -- |g 8.6.2. |t Brownian Formulation -- |g 8.6.3. |t Strassen's Theorem Revisited -- |t Exercises for ʹ 8.6 -- |g ch. 9 |t Convergence of Measures on a Polish Space -- |g 9.1. |t Prohorov -- Varadarajan Theory -- |g 9.1.1. |t Some Background -- |g 9.1.2. |t The Weak Topology -- |g 9.1.3. |t The Levy Metric and Completeness of M1(E) -- |t Exercises for ʹ 9.1 -- |g 9.2. |t Regular Conditional Probability Distributions -- |g 9.2.1. |t Fibering a Measure -- |g 9.2.2. |t Representing Levy Measures via the Ito Map -- |t Exercises for ʹ 9.2 -- |g 9.3. |t Donsker's Invariance Principle -- |g 9.3.1. |t Donsker's Theorem -- |g 9.3.2. |t Rayleigh's Random Flights Model -- |t Exercise for ʹ 9.3 -- |g ch. 10 |t Wiener Measure and Partial Differential Equations -- |g 10.1. |t Martingales and Partial Differential Equations -- |g 10.1.1. |t Localizing and Extending Martingale Representations. |
| 505 | 0 | 0 | |g 10.1.2. |t Minimum Principles -- |g 10.1.3. |t The Hermite Heat Equation -- |g 10.1.4. |t The Arcsine Law -- |g 10.1.5. |t Recurrence and Transience of Brownian Motion -- |t Exercises for ʹ 10.1 -- |g 10.2. |t The Markov Property and Potential Theory -- |g 10.2.1. |t The Markov Property for Wiener Measure -- |g 10.2.2. |t Recurrence in One and Two Dimensions -- |g 10.2.3. |t The Dirichlet Problem -- |t Exercises for ʹ 10.2 -- |g 10.3. |t Other Heat Kernels -- |g 10.3.1. |t A General Construction -- |g 10.3.2. |t The Dirichlet Heat Kernel -- |g 10.3.3. |t Feynman -- Kac Heat Kernels -- |g 10.3.4. |t Ground States and Associated Measures on Pathspace -- |g 10.3.5. |t Producing Ground States -- |t Exercises for ʹ 10.3 -- |g ch. 11 |t Some Classical Potential Theory -- |g 11.1. |t Uniqueness Refined -- |g 11.1.1. |t The Dirichlet Heat Kernel Again -- |g 11.1.2. |t Exiting Through & part;regG -- |g 11.1.3. |t Applications to Questions of Uniqueness -- |g 11.1.4. |t Harmonic Measure -- |t Exercises for ʹ 11.1 -- |g 11.2. |t The Poisson Problem and Green Functions -- |g 11.2.1. |t Green Functions when N & ge; 3. |
| 505 | 0 | 0 | |g 11.2.2. |t Green Functions when N & psi; {1,2} -- |t Exercises for ʹ 11.2 -- |g 11.3. |t Excessive Functions, Potentials, and Riesz Decompositions -- |g 11.3.1. |t Excessive Functions -- |g 11.3.2. |t Potentials and Riesz Decomposition -- |t Exercises for ʹ 11.3 -- |g 11.4. |t Capacity -- |g 11.4.1. |t The Capacitory Potential -- |g 11.4.2. |t The Capacitory Distribution -- |g 11.4.3. |t Wiener's Test -- |g 11.4.4. |t Some Asymptotic Expressions Involving Capacity -- |t Exercises for ʹ 11.4. |
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Datensatz im Suchindex
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| adam_text | |
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| author | Stroock, Daniel W. |
| author_facet | Stroock, Daniel W. |
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| callnumber-search | QA273 .S763 2011eb |
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| contents | Sums of Independent Random Variables -- Independence -- Independent & sigma;-Algebras -- Independent Functions -- The Radomachor Functions -- Exercises for ʹ 1.1 -- The Weak Law of Large Numbers -- Orthogonal Random Variables -- Independent Random Variables -- Approximate Identities -- Exercises for ʹ 1.2 -- Cramer's Theory of Large Deviations -- Exercises for ʹ 1.3 -- The Strong Law of Large Numbers -- Exercises for ʹ 1.4 -- Law of the Iterated Logarithm -- Exercises for ʹ 1.5 -- The Central Limit Theorem -- The Basic Central Limit Theorem -- Lindeberg's Theorem -- Exercises for ʹ 2.1 -- The Berry-Esseen Theorem via Stein's Method -- L1-Berry-Esseen -- The Classical Berry Esseen Theorem -- Exercises for ʹ 2.2. Some Extensions of The Central Limit Theorem -- The Fourier Transform -- Multidimensional Central Limit Theorem -- Higher Moments -- Exercises for ʹ 2.3 -- An Application to Hermite Multipliers -- Hermite Multipliers -- Beckner's Theorem -- Applications of Beckner's Theorem -- Exercises for ʹ 2.4 -- Infinitely Divisible Laws -- Convergence of Measures on RN -- Sequential Compactness in M1RN -- Levy's Continuity Theorem -- Exercises for ʹ 3.1 -- The Levy-Khinchine Formula -- I(RN) Is the Closure of P(RN) -- The Formula -- Exercises for ʹ 3.2 -- Stable Laws -- General Results -- & alpha;-Stable Laws -- Exercises for ʹ 3.3 -- Levy Processes -- Stochastic Processes, Some Generalities -- The Space D(RN) -- Jump Functions -- Exercises for ʹ 4.1 -- Discontinuous Levy Processes -- The Simple Poisson Process. Compound Poisson Processes -- Poisson Jump Processes -- Levy Processes with Bounded Variation -- General, Non-Gaussian Levy Processes -- Exercises for ʹ 4.2 -- Brownian Motion, the Gaussian Levy Process -- Deconstructing Brownian Motion -- Levy's Construction of Brownian Motion -- Levy's Construction in Context -- Brownian Paths Are Non-Differentiable -- General Levy Processes -- Exercises for ʹ 4.3 -- Conditioning and Martingales -- Conditioning -- Kolmogorov's Definition -- Some Extensions -- Exercises for ʹ 5.1 -- Discrete Parameter Martingales -- Doob's Inequality and Marcinkewitz's Theorem -- Doob's Stopping Time Theorem -- Martingale Convergence Theorem -- Reversed Martingales and De Finetti's Theory -- An Application to a Tracking Algorithm -- Exercises for ʹ 5.2 -- Some Extensions and Applications of Martingale Theory. Martingale Theory for a & sigma;-Finite Measure Space -- Banach Space -- Valued Martingales -- Exercises for ʹ 6.1 -- Elements of Ergodic Theory -- The Maximal Ergodic Lemma -- Birkhoff's Ergodic Theorem -- Stationary Sequences -- Continuous Parameter Ergodic Theory -- Exercises for ʹ 6.2 -- Burkholder's Inequality -- Burkholder's Comparison Theorem -- Exercises for ʹ 6.3 -- Continuous Parameter Martingales -- Progressively Measurable Functions -- Martingales: Definition and Examples -- Basic Results -- Stopping Times and Stopping Theorems -- An Integration by Parts Formula -- Exercises for ʹ 7.1 -- Brownian Motion and Martingales -- Levy's Characterization of Brownian Motion -- Doob-Meyer Decomposition, an Easy Case -- Burkholder's Inequality Again -- Exercises for ʹ 7.2. The Reflection Principle Revisited -- Reflecting Symmetric Levy Processes -- Reflected Brownian Motion -- Exercises for ʹ 7.3 -- Gaussian Measures on a Banach Space -- The Classical Wiener Space -- Classical Wiener Measure -- The Classical Cameron -- Martin Space -- Exercises for ʹ 8.1 -- A Structure Theorem for Gaussian Measures -- Fernique's Theorem -- The Basic Structure Theorem -- The Cameron -- Marin Space -- Exercises for ʹ 8.2 -- From Hilbert to Abstract Wiener Space -- An Isomorphism Theorem -- Wiener Series -- Orthogonal Projections -- Pinned Brownian Motion -- Orthogonal Invariance -- Exercises for ʹ 8.3 -- A Large Deviations Result and Strassen's Theorem -- Large Deviations for Abstract Wiener Space -- Strassen's Law of the Iterated Logarithm -- Exercises for ʹ 8.4 -- Euclidean Free Fields -- The Ornstein -- Uhlenbeck Process. Ornstein -- Uhlenbeck as an Abstract Wiener Space -- Higher Dimensional Free Fields -- Exercises for ʹ 8.5 -- Brownian Motion on a Banach Space -- Abstract Wiener Formulation -- Brownian Formulation -- Strassen's Theorem Revisited -- Exercises for ʹ 8.6 -- Convergence of Measures on a Polish Space -- Prohorov -- Varadarajan Theory -- Some Background -- The Weak Topology -- The Levy Metric and Completeness of M1(E) -- Exercises for ʹ 9.1 -- Regular Conditional Probability Distributions -- Fibering a Measure -- Representing Levy Measures via the Ito Map -- Exercises for ʹ 9.2 -- Donsker's Invariance Principle -- Donsker's Theorem -- Rayleigh's Random Flights Model -- Exercise for ʹ 9.3 -- Wiener Measure and Partial Differential Equations -- Martingales and Partial Differential Equations -- Localizing and Extending Martingale Representations. Minimum Principles -- The Hermite Heat Equation -- The Arcsine Law -- Recurrence and Transience of Brownian Motion -- Exercises for ʹ 10.1 -- The Markov Property and Potential Theory -- The Markov Property for Wiener Measure -- Recurrence in One and Two Dimensions -- The Dirichlet Problem -- Exercises for ʹ 10.2 -- Other Heat Kernels -- A General Construction -- The Dirichlet Heat Kernel -- Feynman -- Kac Heat Kernels -- Ground States and Associated Measures on Pathspace -- Producing Ground States -- Exercises for ʹ 10.3 -- Some Classical Potential Theory -- Uniqueness Refined -- The Dirichlet Heat Kernel Again -- Exiting Through & part;regG -- Applications to Questions of Uniqueness -- Harmonic Measure -- Exercises for ʹ 11.1 -- The Poisson Problem and Green Functions -- Green Functions when N & ge; 3. Green Functions when N & psi; {1,2} -- Exercises for ʹ 11.2 -- Excessive Functions, Potentials, and Riesz Decompositions -- Excessive Functions -- Potentials and Riesz Decomposition -- Exercises for ʹ 11.3 -- Capacity -- The Capacitory Potential -- The Capacitory Distribution -- Wiener's Test -- Some Asymptotic Expressions Involving Capacity -- Exercises for ʹ 11.4. |
| ctrlnum | (OCoLC)710993037 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2nd ed. |
| format | Electronic eBook |
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Stroock.</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">2nd ed.</subfield></datafield><datafield tag="260" ind1=" " ind2=" "><subfield code="a">Cambridge ;</subfield><subfield code="a">New York :</subfield><subfield code="b">Cambridge University Press,</subfield><subfield code="c">©2011.</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (xxi, 527 pages)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">"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"--</subfield><subfield code="c">Provided by publisher.</subfield></datafield><datafield tag="504" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references and index.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Print version record.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">Machine generated contents note:</subfield><subfield code="g">ch. 1</subfield><subfield code="t">Sums of Independent Random Variables --</subfield><subfield code="g">1.1.</subfield><subfield code="t">Independence --</subfield><subfield code="g">1.1.1.</subfield><subfield code="t">Independent & sigma;-Algebras --</subfield><subfield code="g">1.1.2.</subfield><subfield code="t">Independent Functions --</subfield><subfield code="g">1.1.3.</subfield><subfield code="t">The Radomachor Functions --</subfield><subfield code="t">Exercises for ʹ 1.1 --</subfield><subfield code="g">1.2.</subfield><subfield code="t">The Weak Law of Large Numbers --</subfield><subfield code="g">1.2.1.</subfield><subfield code="t">Orthogonal Random Variables --</subfield><subfield code="g">1.2.2.</subfield><subfield code="t">Independent Random Variables --</subfield><subfield code="g">1.2.3.</subfield><subfield code="t">Approximate Identities --</subfield><subfield code="t">Exercises for ʹ 1.2 --</subfield><subfield code="g">1.3.</subfield><subfield code="t">Cramer's Theory of Large Deviations --</subfield><subfield code="t">Exercises for ʹ 1.3 --</subfield><subfield code="g">1.4.</subfield><subfield code="t">The Strong Law of Large Numbers --</subfield><subfield code="t">Exercises for ʹ 1.4 --</subfield><subfield code="g">1.5.</subfield><subfield code="t">Law of the Iterated Logarithm --</subfield><subfield code="t">Exercises for ʹ 1.5 --</subfield><subfield code="g">ch. 2</subfield><subfield code="t">The Central Limit Theorem --</subfield><subfield code="g">2.1.</subfield><subfield code="t">The Basic Central Limit Theorem --</subfield><subfield code="g">2.1.1.</subfield><subfield code="t">Lindeberg's Theorem --</subfield><subfield code="g">2.1.2.</subfield><subfield code="t">The Central Limit Theorem --</subfield><subfield code="t">Exercises for ʹ 2.1 --</subfield><subfield code="g">2.2.</subfield><subfield code="t">The Berry-Esseen Theorem via Stein's Method --</subfield><subfield code="g">2.2.1.</subfield><subfield code="t">L1-Berry-Esseen --</subfield><subfield code="g">2.2.2.</subfield><subfield code="t">The Classical Berry Esseen Theorem --</subfield><subfield code="t">Exercises for ʹ 2.2.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">2.3.</subfield><subfield code="t">Some Extensions of The Central Limit Theorem --</subfield><subfield code="g">2.3.1.</subfield><subfield code="t">The Fourier Transform --</subfield><subfield code="g">2.3.2.</subfield><subfield code="t">Multidimensional Central Limit Theorem --</subfield><subfield code="g">2.3.3.</subfield><subfield code="t">Higher Moments --</subfield><subfield code="t">Exercises for ʹ 2.3 --</subfield><subfield code="g">2.4.</subfield><subfield code="t">An Application to Hermite Multipliers --</subfield><subfield code="g">2.4.1.</subfield><subfield code="t">Hermite Multipliers --</subfield><subfield code="g">2.4.2.</subfield><subfield code="t">Beckner's Theorem --</subfield><subfield code="g">2.4.3.</subfield><subfield code="t">Applications of Beckner's Theorem --</subfield><subfield code="t">Exercises for ʹ 2.4 --</subfield><subfield code="g">ch. 3</subfield><subfield code="t">Infinitely Divisible Laws --</subfield><subfield code="g">3.1.</subfield><subfield code="t">Convergence of Measures on RN --</subfield><subfield code="g">3.1.1.</subfield><subfield code="t">Sequential Compactness in M1RN --</subfield><subfield code="g">3.1.2.</subfield><subfield code="t">Levy's Continuity Theorem --</subfield><subfield code="t">Exercises for ʹ 3.1 --</subfield><subfield code="g">3.2.</subfield><subfield code="t">The Levy-Khinchine Formula --</subfield><subfield code="g">3.2.1.</subfield><subfield code="t">I(RN) Is the Closure of P(RN) --</subfield><subfield code="g">3.2.2.</subfield><subfield code="t">The Formula --</subfield><subfield code="t">Exercises for ʹ 3.2 --</subfield><subfield code="g">3.3.</subfield><subfield code="t">Stable Laws --</subfield><subfield code="g">3.3.1.</subfield><subfield code="t">General Results --</subfield><subfield code="g">3.3.2.</subfield><subfield code="t">& alpha;-Stable Laws --</subfield><subfield code="t">Exercises for ʹ 3.3 --</subfield><subfield code="g">ch. 4</subfield><subfield code="t">Levy Processes --</subfield><subfield code="g">4.1.</subfield><subfield code="t">Stochastic Processes, Some Generalities --</subfield><subfield code="g">4.1.1.</subfield><subfield code="t">The Space D(RN) --</subfield><subfield code="g">4.1.2.</subfield><subfield code="t">Jump Functions --</subfield><subfield code="t">Exercises for ʹ 4.1 --</subfield><subfield code="g">4.2.</subfield><subfield code="t">Discontinuous Levy Processes --</subfield><subfield code="g">4.2.1.</subfield><subfield code="t">The Simple Poisson Process.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">4.2.2.</subfield><subfield code="t">Compound Poisson Processes --</subfield><subfield code="g">4.2.3.</subfield><subfield code="t">Poisson Jump Processes --</subfield><subfield code="g">4.2.4.</subfield><subfield code="t">Levy Processes with Bounded Variation --</subfield><subfield code="g">4.2.5.</subfield><subfield code="t">General, Non-Gaussian Levy Processes --</subfield><subfield code="t">Exercises for ʹ 4.2 --</subfield><subfield code="g">4.3.</subfield><subfield code="t">Brownian Motion, the Gaussian Levy Process --</subfield><subfield code="g">4.3.1.</subfield><subfield code="t">Deconstructing Brownian Motion --</subfield><subfield code="g">4.3.2.</subfield><subfield code="t">Levy's Construction of Brownian Motion --</subfield><subfield code="g">4.3.3.</subfield><subfield code="t">Levy's Construction in Context --</subfield><subfield code="g">4.3.4.</subfield><subfield code="t">Brownian Paths Are Non-Differentiable --</subfield><subfield code="g">4.3.5.</subfield><subfield code="t">General Levy Processes --</subfield><subfield code="t">Exercises for ʹ 4.3 --</subfield><subfield code="g">ch. 5</subfield><subfield code="t">Conditioning and Martingales --</subfield><subfield code="g">5.1.</subfield><subfield code="t">Conditioning --</subfield><subfield code="g">5.1.1.</subfield><subfield code="t">Kolmogorov's Definition --</subfield><subfield code="g">5.1.2.</subfield><subfield code="t">Some Extensions --</subfield><subfield code="t">Exercises for ʹ 5.1 --</subfield><subfield code="g">5.2.</subfield><subfield code="t">Discrete Parameter Martingales --</subfield><subfield code="g">5.2.1.</subfield><subfield code="t">Doob's Inequality and Marcinkewitz's Theorem --</subfield><subfield code="g">5.2.2.</subfield><subfield code="t">Doob's Stopping Time Theorem --</subfield><subfield code="g">5.2.3.</subfield><subfield code="t">Martingale Convergence Theorem --</subfield><subfield code="g">5.2.4.</subfield><subfield code="t">Reversed Martingales and De Finetti's Theory --</subfield><subfield code="g">5.2.5.</subfield><subfield code="t">An Application to a Tracking Algorithm --</subfield><subfield code="t">Exercises for ʹ 5.2 --</subfield><subfield code="g">ch. 6</subfield><subfield code="t">Some Extensions and Applications of Martingale Theory.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">6.1.</subfield><subfield code="t">Some Extensions --</subfield><subfield code="g">6.1.1.</subfield><subfield code="t">Martingale Theory for a & sigma;-Finite Measure Space --</subfield><subfield code="g">6.1.2.</subfield><subfield code="t">Banach Space -- Valued Martingales --</subfield><subfield code="t">Exercises for ʹ 6.1 --</subfield><subfield code="g">6.2.</subfield><subfield code="t">Elements of Ergodic Theory --</subfield><subfield code="g">6.2.1.</subfield><subfield code="t">The Maximal Ergodic Lemma --</subfield><subfield code="g">6.2.2.</subfield><subfield code="t">Birkhoff's Ergodic Theorem --</subfield><subfield code="g">6.2.3.</subfield><subfield code="t">Stationary Sequences --</subfield><subfield code="g">6.2.4.</subfield><subfield code="t">Continuous Parameter Ergodic Theory --</subfield><subfield code="t">Exercises for ʹ 6.2 --</subfield><subfield code="g">6.3.</subfield><subfield code="t">Burkholder's Inequality --</subfield><subfield code="g">6.3.1.</subfield><subfield code="t">Burkholder's Comparison Theorem --</subfield><subfield code="g">6.3.2.</subfield><subfield code="t">Burkholder's Inequality --</subfield><subfield code="t">Exercises for ʹ 6.3 --</subfield><subfield code="g">ch. 7</subfield><subfield code="t">Continuous Parameter Martingales --</subfield><subfield code="g">7.1.</subfield><subfield code="t">Continuous Parameter Martingales --</subfield><subfield code="g">7.1.1.</subfield><subfield code="t">Progressively Measurable Functions --</subfield><subfield code="g">7.1.2.</subfield><subfield code="t">Martingales: Definition and Examples --</subfield><subfield code="g">7.1.3.</subfield><subfield code="t">Basic Results --</subfield><subfield code="g">7.1.4.</subfield><subfield code="t">Stopping Times and Stopping Theorems --</subfield><subfield code="g">7.1.5.</subfield><subfield code="t">An Integration by Parts Formula --</subfield><subfield code="t">Exercises for ʹ 7.1 --</subfield><subfield code="g">7.2.</subfield><subfield code="t">Brownian Motion and Martingales --</subfield><subfield code="g">7.2.1.</subfield><subfield code="t">Levy's Characterization of Brownian Motion --</subfield><subfield code="g">7.2.2.</subfield><subfield code="t">Doob-Meyer Decomposition, an Easy Case --</subfield><subfield code="g">7.2.3.</subfield><subfield code="t">Burkholder's Inequality Again --</subfield><subfield code="t">Exercises for ʹ 7.2.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">7.3.</subfield><subfield code="t">The Reflection Principle Revisited --</subfield><subfield code="g">7.3.1.</subfield><subfield code="t">Reflecting Symmetric Levy Processes --</subfield><subfield code="g">7.3.2.</subfield><subfield code="t">Reflected Brownian Motion --</subfield><subfield code="t">Exercises for ʹ 7.3 --</subfield><subfield code="g">ch. 8</subfield><subfield code="t">Gaussian Measures on a Banach Space --</subfield><subfield code="g">8.1.</subfield><subfield code="t">The Classical Wiener Space --</subfield><subfield code="g">8.1.1.</subfield><subfield code="t">Classical Wiener Measure --</subfield><subfield code="g">8.1.2.</subfield><subfield code="t">The Classical Cameron -- Martin Space --</subfield><subfield code="t">Exercises for ʹ 8.1 --</subfield><subfield code="g">8.2.</subfield><subfield code="t">A Structure Theorem for Gaussian Measures --</subfield><subfield code="g">8.2.1.</subfield><subfield code="t">Fernique's Theorem --</subfield><subfield code="g">8.2.2.</subfield><subfield code="t">The Basic Structure Theorem --</subfield><subfield code="g">8.2.3.</subfield><subfield code="t">The Cameron -- Marin Space --</subfield><subfield code="t">Exercises for ʹ 8.2 --</subfield><subfield code="g">8.3.</subfield><subfield code="t">From Hilbert to Abstract Wiener Space --</subfield><subfield code="g">8.3.1.</subfield><subfield code="t">An Isomorphism Theorem --</subfield><subfield code="g">8.3.2.</subfield><subfield code="t">Wiener Series --</subfield><subfield code="g">8.3.3.</subfield><subfield code="t">Orthogonal Projections --</subfield><subfield code="g">8.3.4.</subfield><subfield code="t">Pinned Brownian Motion --</subfield><subfield code="g">8.3.5.</subfield><subfield code="t">Orthogonal Invariance --</subfield><subfield code="t">Exercises for ʹ 8.3 --</subfield><subfield code="g">8.4.</subfield><subfield code="t">A Large Deviations Result and Strassen's Theorem --</subfield><subfield code="g">8.4.1.</subfield><subfield code="t">Large Deviations for Abstract Wiener Space --</subfield><subfield code="g">8.4.2.</subfield><subfield code="t">Strassen's Law of the Iterated Logarithm --</subfield><subfield code="t">Exercises for ʹ 8.4 --</subfield><subfield code="g">8.5.</subfield><subfield code="t">Euclidean Free Fields --</subfield><subfield code="g">8.5.1.</subfield><subfield code="t">The Ornstein -- Uhlenbeck Process.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">8.5.2.</subfield><subfield code="t">Ornstein -- Uhlenbeck as an Abstract Wiener Space --</subfield><subfield code="g">8.5.3.</subfield><subfield code="t">Higher Dimensional Free Fields --</subfield><subfield code="t">Exercises for ʹ 8.5 --</subfield><subfield code="g">8.6.</subfield><subfield code="t">Brownian Motion on a Banach Space --</subfield><subfield code="g">8.6.1.</subfield><subfield code="t">Abstract Wiener Formulation --</subfield><subfield code="g">8.6.2.</subfield><subfield code="t">Brownian Formulation --</subfield><subfield code="g">8.6.3.</subfield><subfield code="t">Strassen's Theorem Revisited --</subfield><subfield code="t">Exercises for ʹ 8.6 --</subfield><subfield code="g">ch. 9</subfield><subfield code="t">Convergence of Measures on a Polish Space --</subfield><subfield code="g">9.1.</subfield><subfield code="t">Prohorov -- Varadarajan Theory --</subfield><subfield code="g">9.1.1.</subfield><subfield code="t">Some Background --</subfield><subfield code="g">9.1.2.</subfield><subfield code="t">The Weak Topology --</subfield><subfield code="g">9.1.3.</subfield><subfield code="t">The Levy Metric and Completeness of M1(E) --</subfield><subfield code="t">Exercises for ʹ 9.1 --</subfield><subfield code="g">9.2.</subfield><subfield code="t">Regular Conditional Probability Distributions --</subfield><subfield code="g">9.2.1.</subfield><subfield code="t">Fibering a Measure --</subfield><subfield code="g">9.2.2.</subfield><subfield code="t">Representing Levy Measures via the Ito Map --</subfield><subfield code="t">Exercises for ʹ 9.2 --</subfield><subfield code="g">9.3.</subfield><subfield code="t">Donsker's Invariance Principle --</subfield><subfield code="g">9.3.1.</subfield><subfield code="t">Donsker's Theorem --</subfield><subfield code="g">9.3.2.</subfield><subfield code="t">Rayleigh's Random Flights Model --</subfield><subfield code="t">Exercise for ʹ 9.3 --</subfield><subfield code="g">ch. 10</subfield><subfield code="t">Wiener Measure and Partial Differential Equations --</subfield><subfield code="g">10.1.</subfield><subfield code="t">Martingales and Partial Differential Equations --</subfield><subfield code="g">10.1.1.</subfield><subfield code="t">Localizing and Extending Martingale Representations.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">10.1.2.</subfield><subfield code="t">Minimum Principles --</subfield><subfield code="g">10.1.3.</subfield><subfield code="t">The Hermite Heat Equation --</subfield><subfield code="g">10.1.4.</subfield><subfield code="t">The Arcsine Law --</subfield><subfield code="g">10.1.5.</subfield><subfield code="t">Recurrence and Transience of Brownian Motion --</subfield><subfield code="t">Exercises for ʹ 10.1 --</subfield><subfield code="g">10.2.</subfield><subfield code="t">The Markov Property and Potential Theory --</subfield><subfield code="g">10.2.1.</subfield><subfield code="t">The Markov Property for Wiener Measure --</subfield><subfield code="g">10.2.2.</subfield><subfield code="t">Recurrence in One and Two Dimensions --</subfield><subfield code="g">10.2.3.</subfield><subfield code="t">The Dirichlet Problem --</subfield><subfield code="t">Exercises for ʹ 10.2 --</subfield><subfield code="g">10.3.</subfield><subfield code="t">Other Heat Kernels --</subfield><subfield code="g">10.3.1.</subfield><subfield code="t">A General Construction --</subfield><subfield code="g">10.3.2.</subfield><subfield code="t">The Dirichlet Heat Kernel --</subfield><subfield code="g">10.3.3.</subfield><subfield code="t">Feynman -- Kac Heat Kernels --</subfield><subfield code="g">10.3.4.</subfield><subfield code="t">Ground States and Associated Measures on Pathspace --</subfield><subfield code="g">10.3.5.</subfield><subfield code="t">Producing Ground States --</subfield><subfield code="t">Exercises for ʹ 10.3 --</subfield><subfield code="g">ch. 11</subfield><subfield code="t">Some Classical Potential Theory --</subfield><subfield code="g">11.1.</subfield><subfield code="t">Uniqueness Refined --</subfield><subfield code="g">11.1.1.</subfield><subfield code="t">The Dirichlet Heat Kernel Again --</subfield><subfield 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| genre | Electronic books. |
| genre_facet | Electronic books. |
| id | ZDB-4-EBA-ocn710993037 |
| illustrated | Not Illustrated |
| indexdate | 2024-10-25T16:18:02Z |
| institution | BVB |
| isbn | 9781139011884 113901188X 9781139011099 113901109X 9780511974243 0511974248 9781139010825 1139010824 |
| language | English |
| lccn | 2010027652 |
| oclc_num | 710993037 |
| open_access_boolean | |
| owner | MAIN |
| owner_facet | MAIN |
| physical | 1 online resource (xxi, 527 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge University Press, |
| record_format | marc |
| spelling | Stroock, Daniel W. Probability theory : an analytic view / Daniel W. Stroock. 2nd ed. Cambridge ; New York : Cambridge University Press, ©2011. 1 online resource (xxi, 527 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier "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. Includes bibliographical references and index. Print version record. 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. 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. 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. 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. 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. 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. 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. 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. Probabilities. http://id.loc.gov/authorities/subjects/sh85107090 Probabilités. probability. aat MATHEMATICS Probability & Statistics General. bisacsh Probabilities fast Wahrscheinlichkeit gnd Wahrscheinlichkeitstheorie gnd http://d-nb.info/gnd/4079013-7 Wahrscheinlichkeit. idszbz Wahrscheinlichkeitstheorie. idszbz Electronic books. has work: Probability theory (Text) https://id.oclc.org/worldcat/entity/E39PCGVwTvmqtCdHBWg9Rgxcvb https://id.oclc.org/worldcat/ontology/hasWork Print version: Stroock, Daniel W. Probability theory. 2nd ed. New York : Cambridge University Press, 2011 9780521761581 (DLC) 2010027652 (OCoLC)649077720 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=357430 Volltext CBO01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=357430 Volltext |
| spellingShingle | Stroock, Daniel W. Probability theory : an analytic view / Sums of Independent Random Variables -- Independence -- Independent & sigma;-Algebras -- Independent Functions -- The Radomachor Functions -- Exercises for ʹ 1.1 -- The Weak Law of Large Numbers -- Orthogonal Random Variables -- Independent Random Variables -- Approximate Identities -- Exercises for ʹ 1.2 -- Cramer's Theory of Large Deviations -- Exercises for ʹ 1.3 -- The Strong Law of Large Numbers -- Exercises for ʹ 1.4 -- Law of the Iterated Logarithm -- Exercises for ʹ 1.5 -- The Central Limit Theorem -- The Basic Central Limit Theorem -- Lindeberg's Theorem -- Exercises for ʹ 2.1 -- The Berry-Esseen Theorem via Stein's Method -- L1-Berry-Esseen -- The Classical Berry Esseen Theorem -- Exercises for ʹ 2.2. Some Extensions of The Central Limit Theorem -- The Fourier Transform -- Multidimensional Central Limit Theorem -- Higher Moments -- Exercises for ʹ 2.3 -- An Application to Hermite Multipliers -- Hermite Multipliers -- Beckner's Theorem -- Applications of Beckner's Theorem -- Exercises for ʹ 2.4 -- Infinitely Divisible Laws -- Convergence of Measures on RN -- Sequential Compactness in M1RN -- Levy's Continuity Theorem -- Exercises for ʹ 3.1 -- The Levy-Khinchine Formula -- I(RN) Is the Closure of P(RN) -- The Formula -- Exercises for ʹ 3.2 -- Stable Laws -- General Results -- & alpha;-Stable Laws -- Exercises for ʹ 3.3 -- Levy Processes -- Stochastic Processes, Some Generalities -- The Space D(RN) -- Jump Functions -- Exercises for ʹ 4.1 -- Discontinuous Levy Processes -- The Simple Poisson Process. Compound Poisson Processes -- Poisson Jump Processes -- Levy Processes with Bounded Variation -- General, Non-Gaussian Levy Processes -- Exercises for ʹ 4.2 -- Brownian Motion, the Gaussian Levy Process -- Deconstructing Brownian Motion -- Levy's Construction of Brownian Motion -- Levy's Construction in Context -- Brownian Paths Are Non-Differentiable -- General Levy Processes -- Exercises for ʹ 4.3 -- Conditioning and Martingales -- Conditioning -- Kolmogorov's Definition -- Some Extensions -- Exercises for ʹ 5.1 -- Discrete Parameter Martingales -- Doob's Inequality and Marcinkewitz's Theorem -- Doob's Stopping Time Theorem -- Martingale Convergence Theorem -- Reversed Martingales and De Finetti's Theory -- An Application to a Tracking Algorithm -- Exercises for ʹ 5.2 -- Some Extensions and Applications of Martingale Theory. Martingale Theory for a & sigma;-Finite Measure Space -- Banach Space -- Valued Martingales -- Exercises for ʹ 6.1 -- Elements of Ergodic Theory -- The Maximal Ergodic Lemma -- Birkhoff's Ergodic Theorem -- Stationary Sequences -- Continuous Parameter Ergodic Theory -- Exercises for ʹ 6.2 -- Burkholder's Inequality -- Burkholder's Comparison Theorem -- Exercises for ʹ 6.3 -- Continuous Parameter Martingales -- Progressively Measurable Functions -- Martingales: Definition and Examples -- Basic Results -- Stopping Times and Stopping Theorems -- An Integration by Parts Formula -- Exercises for ʹ 7.1 -- Brownian Motion and Martingales -- Levy's Characterization of Brownian Motion -- Doob-Meyer Decomposition, an Easy Case -- Burkholder's Inequality Again -- Exercises for ʹ 7.2. The Reflection Principle Revisited -- Reflecting Symmetric Levy Processes -- Reflected Brownian Motion -- Exercises for ʹ 7.3 -- Gaussian Measures on a Banach Space -- The Classical Wiener Space -- Classical Wiener Measure -- The Classical Cameron -- Martin Space -- Exercises for ʹ 8.1 -- A Structure Theorem for Gaussian Measures -- Fernique's Theorem -- The Basic Structure Theorem -- The Cameron -- Marin Space -- Exercises for ʹ 8.2 -- From Hilbert to Abstract Wiener Space -- An Isomorphism Theorem -- Wiener Series -- Orthogonal Projections -- Pinned Brownian Motion -- Orthogonal Invariance -- Exercises for ʹ 8.3 -- A Large Deviations Result and Strassen's Theorem -- Large Deviations for Abstract Wiener Space -- Strassen's Law of the Iterated Logarithm -- Exercises for ʹ 8.4 -- Euclidean Free Fields -- The Ornstein -- Uhlenbeck Process. Ornstein -- Uhlenbeck as an Abstract Wiener Space -- Higher Dimensional Free Fields -- Exercises for ʹ 8.5 -- Brownian Motion on a Banach Space -- Abstract Wiener Formulation -- Brownian Formulation -- Strassen's Theorem Revisited -- Exercises for ʹ 8.6 -- Convergence of Measures on a Polish Space -- Prohorov -- Varadarajan Theory -- Some Background -- The Weak Topology -- The Levy Metric and Completeness of M1(E) -- Exercises for ʹ 9.1 -- Regular Conditional Probability Distributions -- Fibering a Measure -- Representing Levy Measures via the Ito Map -- Exercises for ʹ 9.2 -- Donsker's Invariance Principle -- Donsker's Theorem -- Rayleigh's Random Flights Model -- Exercise for ʹ 9.3 -- Wiener Measure and Partial Differential Equations -- Martingales and Partial Differential Equations -- Localizing and Extending Martingale Representations. Minimum Principles -- The Hermite Heat Equation -- The Arcsine Law -- Recurrence and Transience of Brownian Motion -- Exercises for ʹ 10.1 -- The Markov Property and Potential Theory -- The Markov Property for Wiener Measure -- Recurrence in One and Two Dimensions -- The Dirichlet Problem -- Exercises for ʹ 10.2 -- Other Heat Kernels -- A General Construction -- The Dirichlet Heat Kernel -- Feynman -- Kac Heat Kernels -- Ground States and Associated Measures on Pathspace -- Producing Ground States -- Exercises for ʹ 10.3 -- Some Classical Potential Theory -- Uniqueness Refined -- The Dirichlet Heat Kernel Again -- Exiting Through & part;regG -- Applications to Questions of Uniqueness -- Harmonic Measure -- Exercises for ʹ 11.1 -- The Poisson Problem and Green Functions -- Green Functions when N & ge; 3. Green Functions when N & psi; {1,2} -- Exercises for ʹ 11.2 -- Excessive Functions, Potentials, and Riesz Decompositions -- Excessive Functions -- Potentials and Riesz Decomposition -- Exercises for ʹ 11.3 -- Capacity -- The Capacitory Potential -- The Capacitory Distribution -- Wiener's Test -- Some Asymptotic Expressions Involving Capacity -- Exercises for ʹ 11.4. Probabilities. http://id.loc.gov/authorities/subjects/sh85107090 Probabilités. probability. aat MATHEMATICS Probability & Statistics General. bisacsh Probabilities fast Wahrscheinlichkeit gnd Wahrscheinlichkeitstheorie gnd http://d-nb.info/gnd/4079013-7 Wahrscheinlichkeit. idszbz Wahrscheinlichkeitstheorie. idszbz |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85107090 http://d-nb.info/gnd/4079013-7 |
| title | Probability theory : an analytic view / |
| title_alt | Sums of Independent Random Variables -- Independence -- Independent & sigma;-Algebras -- Independent Functions -- The Radomachor Functions -- Exercises for ʹ 1.1 -- The Weak Law of Large Numbers -- Orthogonal Random Variables -- Independent Random Variables -- Approximate Identities -- Exercises for ʹ 1.2 -- Cramer's Theory of Large Deviations -- Exercises for ʹ 1.3 -- The Strong Law of Large Numbers -- Exercises for ʹ 1.4 -- Law of the Iterated Logarithm -- Exercises for ʹ 1.5 -- The Central Limit Theorem -- The Basic Central Limit Theorem -- Lindeberg's Theorem -- Exercises for ʹ 2.1 -- The Berry-Esseen Theorem via Stein's Method -- L1-Berry-Esseen -- The Classical Berry Esseen Theorem -- Exercises for ʹ 2.2. Some Extensions of The Central Limit Theorem -- The Fourier Transform -- Multidimensional Central Limit Theorem -- Higher Moments -- Exercises for ʹ 2.3 -- An Application to Hermite Multipliers -- Hermite Multipliers -- Beckner's Theorem -- Applications of Beckner's Theorem -- Exercises for ʹ 2.4 -- Infinitely Divisible Laws -- Convergence of Measures on RN -- Sequential Compactness in M1RN -- Levy's Continuity Theorem -- Exercises for ʹ 3.1 -- The Levy-Khinchine Formula -- I(RN) Is the Closure of P(RN) -- The Formula -- Exercises for ʹ 3.2 -- Stable Laws -- General Results -- & alpha;-Stable Laws -- Exercises for ʹ 3.3 -- Levy Processes -- Stochastic Processes, Some Generalities -- The Space D(RN) -- Jump Functions -- Exercises for ʹ 4.1 -- Discontinuous Levy Processes -- The Simple Poisson Process. Compound Poisson Processes -- Poisson Jump Processes -- Levy Processes with Bounded Variation -- General, Non-Gaussian Levy Processes -- Exercises for ʹ 4.2 -- Brownian Motion, the Gaussian Levy Process -- Deconstructing Brownian Motion -- Levy's Construction of Brownian Motion -- Levy's Construction in Context -- Brownian Paths Are Non-Differentiable -- General Levy Processes -- Exercises for ʹ 4.3 -- Conditioning and Martingales -- Conditioning -- Kolmogorov's Definition -- Some Extensions -- Exercises for ʹ 5.1 -- Discrete Parameter Martingales -- Doob's Inequality and Marcinkewitz's Theorem -- Doob's Stopping Time Theorem -- Martingale Convergence Theorem -- Reversed Martingales and De Finetti's Theory -- An Application to a Tracking Algorithm -- Exercises for ʹ 5.2 -- Some Extensions and Applications of Martingale Theory. Martingale Theory for a & sigma;-Finite Measure Space -- Banach Space -- Valued Martingales -- Exercises for ʹ 6.1 -- Elements of Ergodic Theory -- The Maximal Ergodic Lemma -- Birkhoff's Ergodic Theorem -- Stationary Sequences -- Continuous Parameter Ergodic Theory -- Exercises for ʹ 6.2 -- Burkholder's Inequality -- Burkholder's Comparison Theorem -- Exercises for ʹ 6.3 -- Continuous Parameter Martingales -- Progressively Measurable Functions -- Martingales: Definition and Examples -- Basic Results -- Stopping Times and Stopping Theorems -- An Integration by Parts Formula -- Exercises for ʹ 7.1 -- Brownian Motion and Martingales -- Levy's Characterization of Brownian Motion -- Doob-Meyer Decomposition, an Easy Case -- Burkholder's Inequality Again -- Exercises for ʹ 7.2. The Reflection Principle Revisited -- Reflecting Symmetric Levy Processes -- Reflected Brownian Motion -- Exercises for ʹ 7.3 -- Gaussian Measures on a Banach Space -- The Classical Wiener Space -- Classical Wiener Measure -- The Classical Cameron -- Martin Space -- Exercises for ʹ 8.1 -- A Structure Theorem for Gaussian Measures -- Fernique's Theorem -- The Basic Structure Theorem -- The Cameron -- Marin Space -- Exercises for ʹ 8.2 -- From Hilbert to Abstract Wiener Space -- An Isomorphism Theorem -- Wiener Series -- Orthogonal Projections -- Pinned Brownian Motion -- Orthogonal Invariance -- Exercises for ʹ 8.3 -- A Large Deviations Result and Strassen's Theorem -- Large Deviations for Abstract Wiener Space -- Strassen's Law of the Iterated Logarithm -- Exercises for ʹ 8.4 -- Euclidean Free Fields -- The Ornstein -- Uhlenbeck Process. Ornstein -- Uhlenbeck as an Abstract Wiener Space -- Higher Dimensional Free Fields -- Exercises for ʹ 8.5 -- Brownian Motion on a Banach Space -- Abstract Wiener Formulation -- Brownian Formulation -- Strassen's Theorem Revisited -- Exercises for ʹ 8.6 -- Convergence of Measures on a Polish Space -- Prohorov -- Varadarajan Theory -- Some Background -- The Weak Topology -- The Levy Metric and Completeness of M1(E) -- Exercises for ʹ 9.1 -- Regular Conditional Probability Distributions -- Fibering a Measure -- Representing Levy Measures via the Ito Map -- Exercises for ʹ 9.2 -- Donsker's Invariance Principle -- Donsker's Theorem -- Rayleigh's Random Flights Model -- Exercise for ʹ 9.3 -- Wiener Measure and Partial Differential Equations -- Martingales and Partial Differential Equations -- Localizing and Extending Martingale Representations. Minimum Principles -- The Hermite Heat Equation -- The Arcsine Law -- Recurrence and Transience of Brownian Motion -- Exercises for ʹ 10.1 -- The Markov Property and Potential Theory -- The Markov Property for Wiener Measure -- Recurrence in One and Two Dimensions -- The Dirichlet Problem -- Exercises for ʹ 10.2 -- Other Heat Kernels -- A General Construction -- The Dirichlet Heat Kernel -- Feynman -- Kac Heat Kernels -- Ground States and Associated Measures on Pathspace -- Producing Ground States -- Exercises for ʹ 10.3 -- Some Classical Potential Theory -- Uniqueness Refined -- The Dirichlet Heat Kernel Again -- Exiting Through & part;regG -- Applications to Questions of Uniqueness -- Harmonic Measure -- Exercises for ʹ 11.1 -- The Poisson Problem and Green Functions -- Green Functions when N & ge; 3. Green Functions when N & psi; {1,2} -- Exercises for ʹ 11.2 -- Excessive Functions, Potentials, and Riesz Decompositions -- Excessive Functions -- Potentials and Riesz Decomposition -- Exercises for ʹ 11.3 -- Capacity -- The Capacitory Potential -- The Capacitory Distribution -- Wiener's Test -- Some Asymptotic Expressions Involving Capacity -- Exercises for ʹ 11.4. |
| title_auth | Probability theory : an analytic view / |
| title_exact_search | Probability theory : an analytic view / |
| title_full | Probability theory : an analytic view / Daniel W. Stroock. |
| title_fullStr | Probability theory : an analytic view / Daniel W. Stroock. |
| title_full_unstemmed | Probability theory : an analytic view / Daniel W. Stroock. |
| title_short | Probability theory : |
| title_sort | probability theory an analytic view |
| title_sub | an analytic view / |
| topic | Probabilities. http://id.loc.gov/authorities/subjects/sh85107090 Probabilités. probability. aat MATHEMATICS Probability & Statistics General. bisacsh Probabilities fast Wahrscheinlichkeit gnd Wahrscheinlichkeitstheorie gnd http://d-nb.info/gnd/4079013-7 Wahrscheinlichkeit. idszbz Wahrscheinlichkeitstheorie. idszbz |
| topic_facet | Probabilities. Probabilités. probability. MATHEMATICS Probability & Statistics General. Probabilities Wahrscheinlichkeit Wahrscheinlichkeitstheorie Wahrscheinlichkeit. Wahrscheinlichkeitstheorie. Electronic books. |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=357430 |
| work_keys_str_mv | AT stroockdanielw probabilitytheoryananalyticview |