This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.
Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix.Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author’s more advanced textbook in the same series (GTM 274).
Author(s): Jean-François Le Gall
Series: Graduate Texts in Mathematics, 295
Publisher: Springer
Year: 2022
Language: English
Pages: 408
City: Cham
Preface
Contents
List of Symbols
Part I Measure Theory
1 Measurable Spaces
1.1 Measurable Sets
1.2 Positive Measures
1.3 Measurable Functions
Operations on Measurable Functions
1.4 Monotone Class
1.5 Exercises
2 Integration of Measurable Functions
2.1 Integration of Nonnegative Functions
2.2 Integrable Functions
2.3 Integrals Depending on a Parameter
2.4 Exercises
3 Construction of Measures
3.1 Outer Measures
3.2 Lebesgue Measure
3.3 Relation with Riemann Integrals
3.4 A Subset of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Which Is Not Measurable
3.5 Finite Measures on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and the Stieltjes Integral
3.6 The Riesz-Markov-Kakutani Representation Theorem
3.7 Exercises
4 Lp Spaces
4.1 Definitions and the Hölder Inequality
4.2 The Banach Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript p Baseline left parenthesis upper E comma script upper A comma mu right parenthesis) /StPNE pdfmark [/StBMC pdfmarkLp(E,A,μ)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
4.3 Density Theorems in Lp Spaces
4.4 The Radon-Nikodym Theorem
4.5 Exercises
5 Product Measures
5.1 Product σ-Fields
5.2 Product Measures
5.3 The Fubini Theorems
5.4 Applications
5.4.1 Integration by Parts
5.4.2 Convolution
5.4.3 The Volume of the Unit Ball
5.5 Exercises
6 Signed Measures
6.1 Definition and Total Variation
6.2 The Jordan Decomposition
6.3 The Duality Between Lp and Lq
6.4 The Riesz-Markov-Kakutani Representation Theorem for Signed Measures
6.5 Exercises
7 Change of Variables
7.1 The Change of Variables Formula
7.2 The Gamma Function
7.3 Lebesgue Measure on the Unit Sphere
7.4 Exercises
Part II Probability Theory
8 Foundations of Probability Theory
8.1 General Definitions
8.1.1 Probability Spaces
8.1.2 Random Variables
8.1.3 Mathematical Expectation
8.1.4 An Example: Bertrand's Paradox
8.1.5 Classical Laws
8.1.6 Distribution Function of a Real Random Variable
8.1.7 The σ-Field Generated by a Random Variable
8.2 Moments of Random Variables
8.2.1 Moments and Variance
8.2.2 Linear Regression
8.2.3 Characteristic Functions
8.2.4 Laplace Transform and Generating Functions
8.3 Exercises
9 Independence
9.1 Independent Events
9.2 Independence for σ-Fields and Random Variables
9.3 The Borel-Cantelli Lemma
9.4 Construction of Independent Sequences
9.5 Sums of Independent Random Variables
9.6 Convolution Semigroups
9.7 The Poisson Process
9.8 Exercises
10 Convergence of Random Variables
10.1 The Different Notions of Convergence
10.2 The Strong Law of Large Numbers
10.3 Convergence in Distribution
10.4 Two Applications
10.4.1 The Convergence of Empirical Measures
10.4.2 The Central Limit Theorem
10.4.3 The Multidimensional Central Limit Theorem
10.5 Exercises
11 Conditioning
11.1 Discrete Conditioning
11.2 The Definition of Conditional Expectation
11.2.1 Integrable Random Variables
11.2.2 Nonnegative Random Variables
11.2.3 The Special Case of Square Integrable Variables
11.3 Specific Properties of the Conditional Expectation
11.4 Evaluation of Conditional Expectation
11.4.1 Discrete Conditioning
11.4.2 Random Variables with a Density
11.4.3 Gaussian Conditioning
11.5 Transition Probabilities and Conditional Distributions
11.6 Exercises
Part III Stochastic Processes
12 Theory of Martingales
12.1 Definitions and Examples
12.2 Stopping Times
12.3 Almost Sure Convergence of Martingales
12.4 Convergence in Lp When p>1
12.5 Uniform Integrability and Martingales
12.6 Optional Stopping Theorems
12.7 Backward Martingales
12.8 Exercises
13 Markov Chains
13.1 Definitions and First Properties
13.2 A Few Examples
13.2.1 Independent Random Variables
13.2.2 Random Walks on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Z Superscript d) /StPNE pdfmark [/StBMC pdfmarkZdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
13.2.3 Simple Random Walk on a Graph
13.2.4 Galton-Watson Branching Processes
13.3 The Canonical Markov Chain
13.4 The Classification of States
13.5 Invariant Measures
13.6 Ergodic Theorems
13.7 Martingales and Markov Chains
13.8 Exercises
14 Brownian Motion
14.1 Brownian Motion as a Limit of Random Walks
14.2 The Construction of Brownian Motion
14.3 The Wiener Measure
14.4 First Properties of Brownian Motion
14.5 The Strong Markov Property
14.6 Harmonic Functions and the Dirichlet Problem
14.7 Harmonic Functions and Brownian Motion
14.8 Exercises
A A Few Facts from Functional Analysis
Normed Linear Spaces and Banach Spaces
Hilbert Spaces
Notes and Suggestions for Further Reading
References
Index
