Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- 1 Background
- 2 Martingales
- 3 Markov Chains
- 4 Networks and Discrete Analysis
- Part II Results and Applications
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
2 - Martingales
from Part I - Tools and Theory
Published online by Cambridge University Press: 16 May 2024
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- 1 Background
- 2 Martingales
- 3 Markov Chains
- 4 Networks and Discrete Analysis
- Part II Results and Applications
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
Summary
This chapter dives into the theory of (discrete time) martingales.The optional stopping theorem and the martingale convergence theorem are proved.These are used to provide some initial results regarding random walks on groups and bounded harmonic functions. Specifically, the random walk on the integer line is shown to be recurrent. Also, it is shown that the space of bounded harmonic functions is either just the constant functions or has infinite dimension.
Keywords
- Type
- Chapter
- Information
- Harmonic Functions and Random Walks on Groups , pp. 50 - 74Publisher: Cambridge University PressPrint publication year: 2024