This book provides the elements of probability and stochastic processes of direct interest to the applied sciences where probabilistic models play an important role, most notably in the information and communications sciences, computer sciences, operations research, and electrical engineering, but also in fields like epidemiology, biology, ecology, physics, and the earth sciences.
The theoretical tools are presented gradually, not deterring the readers with a wall of technicalities before they have the opportunity to understand their relevance in simple situations. In particular, the use of the so-called modern integration theory (the Lebesgue integral) is postponed until the fifth chapter, where it is reviewed in sufficient detail for a rigorous treatment of the topics of interest in the various domains of application listed above.
The treatment, while mathematical, maintains a balance between depth and accessibility that is suitable for theefficient manipulation, based on solid theoretical foundations, of the four most important and ubiquitous categories of probabilistic models:
- Markov chains, which are omnipresent and versatile models in applied probability
- Poisson processes (on the line and in space), occurring in a range of applications from ecology to queuing and mobile communications networks
- Brownian motion, which models fluctuations in the stock market and the "white noise" of physics
- Wide-sense stationary processes, of special importance in signal analysis and design, as well as in the earth sciences.
This book can be used as a text in various ways and at different levels of study. Essentially, it provides the material for a two-semester graduate course on probability and stochastic processes in a department of applied mathematics or for students in departments where stochastic models play an essential role. The progressive introduction of concepts and tools, along with the inclusion of numerous examples, also makes this book well-adapted for self-study.
Author(s): Pierre Brémaud
Series: Texts in Applied Mathematics 77
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2024
Language: English
Pages: 492
City: Cham
Tags: Markov chains, Convergence of random variables, Conditional expectation, Continuous-time stochastic processes, Brownian motion, Wiener process, Stochastic integral, Markov fields, Simulation algorithm, Poisson processes, Wide-sense stationary processes, Martingales
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
