Foundations of Constructive Probability Theory

✍ Scribed by Yuen-Kwok Chan

• Publisher: Cambridge University Press
• Year: 2021
• Tongue: English
• Leaves: 628
• Series: Encyclopedia of Mathematics and its Applications, 177
• Category: Library

✦ Synopsis

Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.

✦ Table of Contents

Cover
Half-title
Series information
Title page
Copyright information
Dedication
Contents
Acknowledgments
Nomenclature
Part I Introduction and Preliminaries
1 Introduction
2 Preliminaries
2.1 Natural Numbers
2.2 Calculation and Theorem
2.3 Proof by Contradiction
2.4 Recognizing Nonconstructive Theorems
2.5 Prior Knowledge
2.6 Notations and Conventions
3 Partition of Unity
3.1 Abundance of Compact Subsets
3.2 Binary Approximation
3.3 Partition of Unity
3.4 One-Point Compactification
Part II Probability Theory
4 Integration and Measure
4.1 Riemann–Stieljes Integral
4.2 Integration on a Locally Compact Metric Space
4.3 Integration Space: The Daniell Integral
4.4 Complete Extension of Integration
4.5 Integrable Set
4.6 Abundance of Integrable Sets
4.7 Uniform Integrability
4.8 Measurable Function and Measurable Set
4.9 Convergence of Measurable Functions
4.10 Product Integration and Fubini’s Theorem
5 Probability Space
5.1 Random Variable
5.2 Probability Distribution on Metric Space
5.3 Weak Convergence of Distributions
5.4 Probability Density Function and Distribution Function
5.5 Skorokhod Representation
5.6 Independence and Conditional Expectation
5.7 Normal Distribution
5.8 Characteristic Function
5.9 Central Limit Theorem
Part III Stochastic Process
6 Random Field and Stochastic Process
6.1 Random Field and Finite Joint Distributions
6.2 Consistent Family of f.j.d.’s
6.3 Daniell–Kolmogorov Extension
6.4 Daniell–Kolmogorov–Skorokhod Extension
7 Measurable Random Field
7.1 Measurable r.f. That Is Continuous in Probability
7.2 Measurable Gaussian Random Field
8 Martingale
8.1 Filtration and Stopping Time
8.2 Martingale
8.3 Convexity and Martingale Convergence
8.4 Strong Law of Large Numbers
9 a.u. Continuous Process
9.1 Extension from Dyadic Rational Parameters to Real Parameters
9.2 C-Regular Family of f.j.d.’s and C-Regular Process
9.3 a.u. Hoelder Process
9.4 Brownian Motion
9.5 a.u. Continuous Gaussian Process
10 a.u. Càdlàg Process
10.1 Càdlàg Function
10.2 Skorokhod Space D[0,1] of Càdlàg Functions
10.3 a.u. Càdlàg Process
10.4 D-Regular Family of f.j.d.’s and D-Regular Process
10.5 Right-Limit Extension of D-Regular Process Is a.u. Càdlàg
10.6 Continuity of the Right-Limit Extension
10.7 Strong Right Continuity in Probability
10.8 Sufficient Condition for an a.u. Càdlàg Martingale
10.9 Sufficient Condition for Right-Hoelder Process
10.10 a.u. Càdlàg Process on [0,∞)
10.11 First Exit Time for a.u. Càdlàg Process
11 Markov Process
11.1 Markov Process and Strong Markov Process
11.2 Transition Distribution
11.3 Markov Semigroup
11.4 Markov Transition f.j.d.’s
11.5 Construction of a Markov Process from a Semigroup
11.6 Continuity of Construction
11.7 Feller Semigroup and Feller Process
11.8 Feller Process Is Strongly Markov
11.9 Abundance of First Exit Times
11.10 First Exit Time for Brownian Motion
Appendices
Appendix A Change of Integration Variables
Appendix B Taylor's Theorem
References
Index