دانلود کتاب Probability: An Introduction Through Theory and Exercises (Universitext)

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Preface
Contents
Notation
1 Elements of Measure Theory
1.1 Measurable Spaces, Measurable Functions
1.2 Real Measurable Functions
1.3 Measures
1.4 Integration
1.5 Important Examples
1.6 Lp Spaces
1.7 Product Spaces, Product Measures
Exercises
2 Probability
2.1 Random Variables, Laws, Expectation
2.2 Independence
2.3 Computation of Laws
2.4 A Convexity Inequality: Jensen
2.5 Moments, Variance, Covariance
2.6 Characteristic Functions
2.7 The Laplace Transform
2.8 Multivariate Gaussian Laws
2.9 Quadratic Functionals of Gaussian r.v.\'s, a Bit of Statistics
Exercises
3 Convergence
3.1 Convergence of r.v.\'s
3.2 Almost Sure Convergence and the Borel-Cantelli Lemma
3.3 Strong Laws of Large Numbers
3.4 Weak Convergence of Measures
3.5 Convergence in Law
3.6 Uniform Integrability
3.7 Convergence in a Gaussian World
3.8 The Central Limit Theorem
3.9 Application: Pearson\'s Theorem, the χ2 Test
3.10 Some Useful Complements
Exercises
4 Conditioning
4.1 Introduction
4.2 Conditional Expectation
4.3 Conditional Laws
4.4 The Conditional Laws of Gaussian Vectors
Exercises
5 Martingales
5.1 Stochastic Processes
5.2 Martingales: Definitions and General Facts
5.3 Doob\'s Decomposition
5.4 Stopping Times
5.5 The Stopping Theorem
5.6 Almost Sure Convergence
5.7 Doob\'s Inequality and Lp Convergence, p>1
5.8 L1 Convergence, Regularity
Exercises
6 Complements
6.1 Random Number Generation, Simulation
6.2 Tightness and the Topology of Weak Convergence
6.3 Applications
Exercises
7 Solutions
References
Index