An Introduction to Applied Probability

Preface
	Part I: The Elementary Calculus
	Part II: The Essential Theory
	Part III: The Important Models
Contents
Chapter 1 Basic Notions
	1.1 Outcomes and Events
	1.2 Probability of Events
	1.3 Independence and Conditioning
	1.4 Counting Models
	1.5 Exercises
Chapter 2 Discrete Random Variables
	2.1 Probability Distribution and Expectation
		Independence and Conditional Independence
		Expectation
		Markov’s Inequality
		Jensen’s Inequality
		Moment Bounds
		Product Rule for Expectation
	2.2 Remarkable Discrete Distributions
		Uniform
		Binomial
		Geometric
		Poisson
		Hypergeometric
		Multinomial
	2.3 Generating Functions
		Moments from the Generating Function
		Random Sums
		Branching Trees
	2.4 Conditional Expectation I
	2.5 Exercises
Chapter 3 Continuous Random Vectors
	3.1 Random Variables with Real Values
		Expectation
		Mean and Variance
		Remarkable Continuous Random Variables
		Characteristic Functions
		Laplace Transforms
		Random Vectors
	3.2 Continuous Random Vectors
		Product Formula for Expectations
		Freeze and Integrate
		Characteristic Functions and Laplace Transforms of Random Vectors
		Characteristic Function Test for Independence
		Random Sums and Wald’s Identity
		Smooth Change of Variables
		Order Statistics
		Sampling a Distribution
	3.3 Square-integrable Random Variables
		Inner Product and Schwarz’s Inequality
		The Correlation Coefficient
		Covariance Matrices
		Linear Regression
	3.4 Gaussian Vectors
		Mixed Moments of Gaussian Vectors
		Independence and Non-Correlation
		Probability Density of a Non-degenerate Gaussian Vector
		Empirical Mean and Variance of the Gaussian Distribution
	3.5 Conditional Expectation II
		Properties of the Conditional Expectation
		Bayesian Tests of Hypotheses
	3.6 Exercises
Chapter 4 The Lebesgue Integral
	4.1 Measurable Functions and Measures
		Measurable Functions
		Measure
		μ-negligible sets
		Cumulative Distribution Function
		Caratheodory’s Theorem
	4.2 The Integral
	4.3 Basic Properties of the Integral
		Beppo Levi, Fatou and Lebesgue
		Differentiation under the Integral Sign
	4.4 The Big Theorems
		The Image Measure Theorem
		The Radon–Nikod´ym Theorem
		The Fubini–Tonelli Theorem
		The Formula of Integration by Parts
		Lp-spaces and the Riesz–Fischer Theorem
	4.5 Exercises
	4.6 Solutions
Chapter 5 From Integral to Expectation
	5.1 Translation
	5.2 The Distribution of a Random Element
	5.3 Characteristic Functions
	5.4 Independence
		The Product Formula
	5.5 Conditional Expectation III
	5.6 General Theory of Conditional Expectation
		A Special Case
		Properties of the Conditional Expectation
		The L2-theory of Conditional Expectation
		Nonlinear Regression
	5.7 Exercises
	5.8 Solutions
Chapter 6 Convergence Almost Sure
	6.1 A Sufficient Condition and a Criterion
		The Borel–Cantelli Lemma
		A Sufficient Condition
		A Criterion
	6.2 The Strong Law of Large Numbers
		Kolmogorov’s Strong Law of Large Numbers
		Large Deviations from the Strong Law of Large Numbers
	6.3 Kolmogorov’s Zero-one Law
	6.4 Related Types of Convergence
		Convergence in Probability
		Convergence in the Quadratic Mean
	6.5 Uniform Integrability
	6.6 Exercises
Chapter 7 Convergence in Distribution
	7.1 Paul Lévy’s Criterion
		Bochner’s Theorem
	7.2 The Central Limit Theorem
		Confidence Intervals
	7.3 Convergence in Variation
	7.4 The Rank of Convergence in Distribution
		A Stability Property of the Gaussian Distribution
		Skorokhod’s Theorem
	7.5 Exercises
Chapter 8 Martingales
	8.1 The Martingale Property
		Convex Functions of Martingales
		Martingale Transforms and Stopped Martingales
	8.2 Martingale Inequalities
		Kolmogorov’s Inequality
		Doob’s Inequality
		Hoeffding’s Inequality
	8.3 The Optional Sampling Theorem
		Wald’s Formulas
	8.4 The Martingale Convergence Theorem
		The Upcrossing Inequality
		Backwards (or Reverse) Martingales
		The Robbins–Sigmund Theorem
	8.5 Square-integrable Martingales
		Doob’s decomposition
		The Martingale Law of Large Numbers
	8.6 Exercises
Chapter 9 Markov Chains
	9.1 The Transition Matrix
		First-step Analysis
		Communication and Period
		Stationary Distributions
		Reversible Chains
		The Strong Markov Property
		The Cycle Independence Property
	9.2 Recurrence
		The Potential Matrix Criterion
		Invariant Measure
		The Stationary Distribution Criterion of Positive Recurrence
		Birth-and-Death Markov Chain
		Foster’s Theorem
	9.3 Long-run Behavior
		The Markov Chain Ergodic Theorem
		The Markov Chain Convergence Theorem
	9.4 Absorption
		Before Absorption
		Time to Absorption
		Final Destination
	9.5 The Markov Property on Graphs
		Gibbs Distributions
		The Hammersley–Clifford Theorem
	9.6 Monte Carlo Markov Chains
		Simulation of Random Fields
		The Propp–Wilson Algorithm
	9.7 Exercises
Chapter 10 Poisson Processes
	10.1 Poisson Processes on the Line
		The Counting Process of an HPP
		Competing Poisson Processes
	10.2 Generalities on Point Processes
		Independent Point Processes
		Marked Point Processes
		Point Process Integrals
		The Intensity Measure
		Campbell’s Formula
		The Laplace Functional
	10.3 Spatial Poisson Processes
		Doubly Stochastic Poisson Processes
		The Covariance Formula
		The Exponential Formula
		Marked Spatial Poisson Processes
	10.4 Operations on Poisson Processes
		Thinning and Coloring
		Transportation
		Poisson Shot Noise
	10.5 Exercises
Chapter 11 Brownian Motion
	11.1 Continuous-time Stochastic Processes
		Second-order Stochastic Processes
		Wide-sense Stationarity
	11.2 Gaussian Processes
		The Wiener Process
		Pathology
		The Brownian Bridge
		Gauss–Markov Processes
	11.3 The Wiener–Doob Integral
		Gaussian Subspaces
		Construction of the Wiener–Doob Integral
		A Formula of Integration by Parts Theorem 11.3.6
	11.4 Two Applications
		Langevin’s Equation
		The Cameron–Martin Formula
	11.5 Fractal Brownian Motion
	11.6 Exercises
Chapter 12 Wide-sense Stationary Processes
	12.1 The Power Spectral Measure
		The General Case
		Special Cases
	12.2 Filtering of WSS Sochastic Processes
		White Noise
	12.3 The Cramér–Khinchin Decomposition
		A Plancherel–Parseval Formula
		Linear Operations on WSS Stochastic Processes
		Stochastic Processes
		Linear Transformations of Gaussian Processes
	12.4 Multivariate WSS Stochastic Processes
		Band-pass Stochastic Processes
	12.5 Exercises
Appendix A: A Review of Hilbert Spaces
	Basic Definitions
	Schwarz’s Inequality
	Isometric Extension
	Orthogonal Projection
Bibliography
Index