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Probability Theory and Examples

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ISBN-10: 0521765390

ISBN-13: 9780521765398

Edition: 4th 2010

Authors: Rick Durrett

List price: $100.95
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Book details

List price: $100.95
Edition: 4th
Copyright year: 2010
Publisher: Cambridge University Press
Publication date: 8/30/2010
Binding: Hardcover
Pages: 440
Size: 7.00" wide x 10.00" long x 1.00" tall
Weight: 2.332
Language: English

Preface
Measure Theory
Probability Spaces
Distributions
Random Variables
Integration
Properties of the Integral
Expected Value
Inequalities
Integration to the Limit
Computing Expected Values
Product Measures, Fubini's Theorem
Laws of Large Numbers
Independence
Sufficient Conditions for Independence
Independence, Distribution, and Expectation
Sums of Independent Random Variables
Constructing Independent Random Variables
Weak Laws of Large Numbers
L<sup>2</sup> Weak Laws
Triangular Arrays
Truncation
Borel-Cantelli Lemmas
Strong Law of Large Numbers
Convergence of Random Series<sup>*</sup>
Rates of Convergence
Infinite Mean
Large Deviations<sup>*</sup>
Central Limit Theorems
The De Moivre-Laplace Theorem
Weak Convergence
Examples
Theory
Characteristic Functions
Definition, Inversion Formula
Weak Convergence
Moments and Derivatives
Polya's Criterion<sup>*</sup>
The Moment Problem<sup>*</sup>
Central Limit Theorems
i.i.d. Sequences
Triangular Arrays
Prime Divisors (Erd�s-Kac)<sup>*</sup>
Rates of Convergence (Berry-Esseen)<sup>*</sup>
Local Limit Theorems<sup>*</sup>
Poisson Convergence
The Basic Limit Theorem
Two Examples with Dependence
Poisson Processes
Stable Laws<sup>*</sup>
Infinitely Divisible Distributions<sup>*</sup>
Limit Theorems in R<sup>d</sup>
Random Walks
Stopping Times
Recurrence
Visits to 0, Arcsine Laws<sup>*</sup>
Renewal Theory<sup>*</sup>
Martingales
Conditional Expectation
Examples
Properties
Regular Conditional Probabilities<sup>*</sup>
Martingales, Almost Sure Convergence
Examples
Bounded Increments
Polya's Urn Scheme
Radon-Nikodym Derivatives
Branching Processes
Doob's Inequality, Convergence in L<sup>p</sup>
Square Integrable Martingales<sup>*</sup>
Uniform Integrability, Convergence in L<sup>1</sup>
Backwards Martingales
Optional Stopping Theorems
Markov Chains
Definitions
Examples
Extensions of the Markov Property
Recurrence and Transience
Stationary Measures
Asymptotic Behavior
Periodicity, Tail �-field<sup>*</sup>
General State Space<sup>*</sup>
Recurrence and Transience
Stationary Measures
Convergence Theorem
GI/G/1 Queue
Ergodic Theorems
Definitions and Examples
Birkhoff's Ergodic Theorem
Recurrence
A Subadditive Ergodic Theorem<sup>*</sup>
Applications<sup>*</sup>
Brownian Motion
Definition and Construction
Markov Property, Blumenthal's 0-1 Law
Stopping Times, Strong Markov Property
Path Properties
Zeros of Brownian Motion
Hitting Times
L�vy's Modulus of Continuity
Martingales
Multidimensional Brownian Motion
Donsker's Theorem
Empirical Distributions, Brownian Bridge
Laws of the Iterated Logarithm<sup>*</sup>
Appendix A: Measure Theory Details
Carath�odory's Extension Theorem
Which Sets Are Measurable?
Kolmogorov's Extension Theorem
Radon-Nikodym Theorem
Differentiating under the Integral
References
Index