We became familiar with martingales X = (X n )n∈N0 as fair games and found that under certain transformations (optional stopping, discrete stochastic integral) martingales turn into martingales. In this chapter, we will see that under weak conditions (non-negativity or uniform integrability) martingales converge almost surely. Furthermore, the martingale structure implies Lp-convergence under assumptions that are (formally) weaker than those of Chapter 7. The basic ideas of this chapter are Doob’s inequality (Theorem 11.2) and the upcrossing inequality (Lemma 11.3).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2008 Springer-Verlag London Limited
About this chapter
Cite this chapter
(2008). Martingale Convergence Theorems and Their Applications. In: Probability Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-84800-048-3_11
Download citation
DOI: https://doi.org/10.1007/978-1-84800-048-3_11
Publisher Name: Springer, London
Print ISBN: 978-1-84800-047-6
Online ISBN: 978-1-84800-048-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)