Probability Theory and Examples (Fifth Edition) - Rick Durrett (англ.)

Probability Theory and Examples (Fifth Edition) Рика Дарретта - это классический академический учебник по теории вероятностей, широко используемый в ведущих университетах мира. Книга предлагает строгий и последовательный курс теории вероятностей с упором на примеры, упражнения и практические применения.

Издание охватывает законы больших чисел, центральную предельную теорему, мартингалы, цепи Маркова, эргодические теоремы, случайные блуждания и броуновское движение. Автор следует принципу обучения через примеры - в книге более 200 иллюстративных примеров и около 450 задач различной сложности.

Пятое издание обновлено с учетом современных приложений в статистике, стохастических процессах, финансовой математике и прикладных науках. Учебник предназначен для студентов старших курсов, аспирантов, исследователей и специалистов, которым требуется глубокое и строгое понимание вероятностных методов.

Preface

1.1 Probability Spaces
1.2 Distributions
1.3 Random Variables
1.4 Integration
1.5 Properties of the Integral
1.6 Expected Value
 1.6.1 Inequalities
 1.6.2 Integration to the Limit
 1.6.3 Computing Expected Values
1.7 Product Measures, Fubini’s Theorem

2.1 Independence
 2.1.1 Sufficient Conditions for Independence
 2.1.2 Independence, Distribution, and Expectation
 2.1.3 Sums of Independent Random Variables
 2.1.4 Constructing Independent Random Variables
2.2 Weak Laws of Large Numbers
 2.2.1 L² Weak Laws
 2.2.2 Triangular Arrays
 2.2.3 Truncation
2.3 Borel-Cantelli Lemmas
2.4 Strong Law of Large Numbers
2.5 Convergence of Random Series*
 2.5.1 Rates of Convergence
 2.5.2 Infinite Mean
2.6 Renewal Theory*
2.7 Large Deviations*

3.1 The De Moivre-Laplace Theorem
3.2 Weak Convergence
 3.2.1 Examples
 3.2.2 Theory
3.3 Characteristic Functions
 3.3.1 Definition, Inversion Formula
 3.3.2 Weak Convergence
 3.3.3 Moments and Derivatives
 3.3.4 Polya’s Criterion*
 3.3.5 The Moment Problem*
3.4 Central Limit Theorems
 3.4.1 i.i.d. Sequences
 3.4.2 Triangular Arrays
 3.4.3 Prime Divisors (Erdos-Kac)*
 3.4.4 Rates of Convergence (Berry-Esseen)*
3.5 Local Limit Theorems*
3.6 Poisson Convergence
 3.6.1 The Basic Limit Theorem
 3.6.2 Two Examples with Dependence
3.7 Poisson Processes
 3.7.1 Compound Poisson Processes
 3.7.2 Thinning
 3.7.3 Conditioning
3.8 Stable Laws*
3.9 Infinitely Divisible Distributions*
3.10 Limit Theorems in Rᵈ

4.1 Conditional Expectation
 4.1.1 Examples
 4.1.2 Properties
 4.1.3 Regular Conditional Probabilities*
4.2 Martingales, Almost Sure Convergence
4.3 Examples
 4.3.1 Bounded Increments
 4.3.2 Polya’s Urn Scheme
 4.3.3 Radon-Nikodym Derivatives
 4.3.4 Branching Processes
4.4 Doob’s Inequality, Convergence in Lᵖ, p > 1
4.5 Square Integrable Martingales*
4.6 Uniform Integrability, Convergence in L¹
4.7 Backwards Martingales
4.8 Optional Stopping Theorems
 4.8.1 Applications to Random Walks
4.9 Combinatorics of Simple Random Walk*

5.1 Examples
5.2 Construction, Markov Properties
5.3 Recurrence and Transience
5.4 Recurrence of Random Walks*
5.5 Stationary Measures
5.6 Asymptotic Behavior
5.7 Periodicity, Tail σ-Field*
5.8 General State Space*
 5.8.1 Recurrence and Transience
 5.8.2 Stationary Measures
 5.8.3 Convergence Theorem
 5.8.4 GI/G/1 Queue

6.1 Definitions and Examples
6.2 Birkhoff’s Ergodic Theorem
6.3 Recurrence
6.4 A Subadditive Ergodic Theorem
6.5 Applications

7.1 Definition and Construction
7.2 Markov Property, Blumenthal’s 0-1 Law
7.3 Stopping Times, Strong Markov Property
7.4 Path Properties
 7.4.1 Zeros of Brownian Motion
 7.4.2 Hitting Times
7.5 Martingales
7.6 Ito’s Formula*

8.1 Donsker’s Theorem
8.2 CLTs for Martingales
8.3 CLTs for Stationary Sequences
 8.3.1 Mixing Properties
8.4 Empirical Distributions, Brownian Bridge
8.5 Laws of the Iterated Logarithm

9.1 Martingales
9.2 Heat Equation
9.3 Inhomogeneous Heat Equation
9.4 Feynman-Kac Formula
9.5 Dirichlet Problem
 9.5.1 Exit Distributions
9.6 Green’s Functions and Potential Kernels
9.7 Poisson’s Equation
 9.7.1 Occupation Times
9.8 Schrodinger Equation

Appendix A. Measure Theory Details
A.1 Caratheodory’s Extension Theorem
A.2 Which Sets Are Measurable?
A.3 Kolmogorov’s Extension Theorem
A.4 Radon-Nikodym Theorem
A.5 Differentiating under the Integral

References