Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Sums of Independent Random Variables
- 2 The Central Limit Theorem
- 3 Infinitely Divisible Laws
- 4 Lévy Processes
- 5 Conditioning and Martingales
- 6 Some Extensions and Applications of Martingale Theory
- 7 Continuous Parameter Martingales
- 8 Gaussian Measures on a Banach Space
- 9 Convergence of Measures on a Polish Space
- 10 Wiener Measure and Partial Differential Equations
- 11 Some Classical Potential Theory
- References
- Index
6 - Some Extensions and Applications of Martingale Theory
Published online by Cambridge University Press: 07 November 2024
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Sums of Independent Random Variables
- 2 The Central Limit Theorem
- 3 Infinitely Divisible Laws
- 4 Lévy Processes
- 5 Conditioning and Martingales
- 6 Some Extensions and Applications of Martingale Theory
- 7 Continuous Parameter Martingales
- 8 Gaussian Measures on a Banach Space
- 9 Convergence of Measures on a Polish Space
- 10 Wiener Measure and Partial Differential Equations
- 11 Some Classical Potential Theory
- References
- Index
Summary
Chapter 6 opens with extensions of martingale theory in two directions: to σ-finite measures and to random variables with values in a Banach space. In §6.2 I prove Burkholder’s Inequality for martingales with values in a Hilbert space. The derivation that I give is essentially the same as Burkholder’s second proof, the one that gives optimal constants. Finally, the results in §6.1 are used in §6.3 to derive Birkhoff’s Individual Ergodic Theorem and a couple of its applications.
- Type
- Chapter
- Information
- Probability Theory, An Analytic View , pp. 200 - 234Publisher: Cambridge University PressPrint publication year: 2024