Book contents
- Frontmatter
- Contents
- Prelude
- Dependence chart
- 1 Prologue
- 2 The pleasures of counting
- 3 σ-algebras
- 4 Measures
- 5 Uniqueness of measures
- 6 Existence of measures
- 7 Measurable mappings
- 8 Measurable functions
- 9 Integration of positive functions
- 10 Integrals of measurable functions and null sets
- 11 Convergence theorems and their applications
- 12 The function spaces Lp, 1 ≤ p ≤ ∞
- 13 Product measures and Fubini's theorem
- 14 Integrals with respect to image measures
- 15 Integrals of images and Jacobi's transformation rule
- 16 Uniform integrability and Vitali's convergence theorem
- 17 Martingales
- 18 Martingale convergence theorems
- 19 The Radon–Nikodým theorem and other applications of martingales
- 20 Inner product spaces
- 21 Hilbert space h
- 22 Conditional expectations in L2
- 23 Conditional expectations in Lp
- 24 Orthonormal systems and their convergence behaviour
- Appendix A lim inf and lim sup
- Appendix B Some facts from point-set topology
- Appendix C The volume of a parallelepiped
- Appendix D Non-measurable sets
- Appendix E A summary of the Riemann integral
- Further reading
- References
- Notation index
- Name and subject index
17 - Martingales
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Prelude
- Dependence chart
- 1 Prologue
- 2 The pleasures of counting
- 3 σ-algebras
- 4 Measures
- 5 Uniqueness of measures
- 6 Existence of measures
- 7 Measurable mappings
- 8 Measurable functions
- 9 Integration of positive functions
- 10 Integrals of measurable functions and null sets
- 11 Convergence theorems and their applications
- 12 The function spaces Lp, 1 ≤ p ≤ ∞
- 13 Product measures and Fubini's theorem
- 14 Integrals with respect to image measures
- 15 Integrals of images and Jacobi's transformation rule
- 16 Uniform integrability and Vitali's convergence theorem
- 17 Martingales
- 18 Martingale convergence theorems
- 19 The Radon–Nikodým theorem and other applications of martingales
- 20 Inner product spaces
- 21 Hilbert space h
- 22 Conditional expectations in L2
- 23 Conditional expectations in Lp
- 24 Orthonormal systems and their convergence behaviour
- Appendix A lim inf and lim sup
- Appendix B Some facts from point-set topology
- Appendix C The volume of a parallelepiped
- Appendix D Non-measurable sets
- Appendix E A summary of the Riemann integral
- Further reading
- References
- Notation index
- Name and subject index
Summary
Martingales are a key tool of modern probability theory, in particular, when it comes to a.e. convergence assertions and related limit theorems. The origins of martingale techniques can be traced back to analysis papers by Kac, Marcinkiewicz, Paley, Steinhaus, Wiener and Zygmund from the early 1930s on independent (or orthogonal) functions and the convergence of certain series of functions, see e.g. the paper by Marcinkiewicz and Zygmund which contains many references. The theory of martingales as we know it now goes back to Doob and most of the material of this and the following chapter can be found in his seminal monograph from 1953.
We want to understand martingales as an analysis tool which will be useful for the study of Lp- and almost everywhere convergence and, in particular, for the further development of measure and integration theory. Our presentation differs somewhat from the standard way to introduce martingales – conditional expectations will be defined later in Chapter 22 – but the results and their proofs are pretty much the usual ones. The only difference is that we develop the theory for σ-finite measure spaces rather than just for probability spaces. Those readers who are familiar with martingales and the language of conditional expectations we ask for patience until Chapter 23, in particular Theorem 23.9, when we catch up with these notions.
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- Chapter
- Information
- Measures, Integrals and Martingales , pp. 176 - 189Publisher: Cambridge University PressPrint publication year: 2005