Probability Theory An Analytic View 2nd Edition by Daniel Stroock 0521761581 9780521761581

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ISBN 10: 0521761581

ISBN 13: 9780521761581

Author: Daniel W. Stroock

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.

Probability Theory An Analytic View 2nd Table of contents:

Chapter 1 Sums of Independent Random Variables

1.1 Independence

1.1.1. Independent sigma-Algebras

1.1.2. Independent Functions

1.1.3. The Rademacher Functions

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

1.3 Cramer’s Theory of Large Deviations

1.4 The Strong Law of Large Numbers

1.5 Law of the Iterated Logarithm

Chapter 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

2.2 The Berry–Esseen Theorem via Stein’s Method

2.2.1. L1-Berry–Esseen

2.2.2. The Classical Berry–Esseen Theorem

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

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

Chapter 3 In nitely Divisible Laws

3.1 Convergence of Measures on RN

3.1.1. Sequential Compactness in M1(RN)

3.1.2. Levy’s Continuity Theorem

3.2 The Levy–Khinchine Formula

3.3 Stable Laws

3.3.1. General Results

3.3.2. alpha-Stable Laws

Chapter 4 Levy Processes

4.1 Stochastic Processes, Some Generalities

4.1.1. The Space

4.1.2. Jump Functions

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

4.3 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

Chapter 5 Conditioning and Martingales

5.1 Conditioning

5.1.1. Kolmogorov’s Definition

5.1.2. Some Extensions

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

Chapter 6 Some Extensions and Applications of Martingale Theory

6.1 Some Extensions

6.1.1. Martingale Theory for sigma-Finite Measure Space

6.1.2. Banach Space–Valued Martingales

6.2 Elements of Ergodic Theory

6.2.1. The Maximal Ergodic Lemma

6.2.2. Birkho’s Ergodic Theorem

6.2.3. Stationary Sequences

6.2.4. Continuous Parameter Ergodic Theory

6.3 Burkholder’s Inequality

6.3.1. Burkholder’s Comparison Theorem

6.3.2. Burkholder’s Inequality

Chapter 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

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

7.3 The Reection Principle Revisited

7.3.1. Reecting Symmetric Levy Processes

7.3.2. Reected Brownian Motion

Chapter 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

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

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

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

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

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

Chapter 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)

9.2 Regular Conditional Probability Distributions

9.2.1. Fibering a Measure

9.2.2. Representing Levy Measures via the Ito Map

9.3 Donsker’s Invariance Principle

9.3.1. Donsker’s Theorem

9.3.2. Rayleigh’s Random Flights Model

Chapter 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

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

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

Chapter 11 Some Classical Potential Theory

11.1 Uniqueness Refined

11.1.1. The Dirichlet Heat Kernel Again

11.1.2. Exiting Through

11.1.3. Applications to Questions of Uniqueness

11.1.4. Harmonic Measure

11.2 The Poisson Problem and Green Functions

11.2.1. Green Functions when…

11.2.2. Green Functions when…

11.3 Excessive Functions, Potentials, and Riesz Decompositions

11.3.1. Excessive Functions

11.3.2. Potentials and Riesz Decomposition

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

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