Probability: An Introduction Through Theory and Exercises (Universitext)

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