A Little Book of Martingales

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Measure

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   We recollect the essential concepts and results from Measure Theory, mostly without proofs. We state the Monotone class theorem, Carathéodory’s extension theorem, and the Borel–Cantelli lemma. Essential results on integration, such as Fatou’s lemma, Monotone convergence theorem, Dominated convergence theorem, Fubini’s theorem, and some standard inequalities are given without proof. Some of the more important results such as the construction of Lebesgue–Stieltjes measures on \(\mathbb {R}^n\), Jordan-Hahn decomposition, Radon–Nikodym theorem and the Lebesgue decomposition theorem are developed in details. The construction of the Lebesgue–Stieltjes measures is given in some details. Readers who need more details, can refer to one of many well-known books available.

3. Chapter 2. Signed Measure

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   This chapter is on signed measures. The Jordan–Hahn theorem decomposes a signed measure into the difference of two measures in a unique way. This result is crucial to prove the Radon–Nikodym theorem which generalizes the concepts of the derivative and the indefinite integral, to measures. It leads to the Lebesgue decomposition theorem which decomposes any measure on the Borel σ-field into sum of three measures. The three component measures being respectively, (i) absolutely continuous with respect to the Lebesgue measure, (ii) purely discrete, and (iii) singular with respect to the Lebesgue measure but without any discrete component.

4. Chapter 3. Conditional Expectation

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   We introduce the notion of conditional expectation which is based on the Radon–Nikodym theorem. We establish its basic properties, including Jensen’s inequality. We also show the existence of a regular conditional probability distribution. The useful notion of uniform integrability is also a part of this chapter.

5. Chapter 4. Martingales

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   We introduce martingales, sub- and super-martingales. We define stopping times, which is fundamental in martingale theory, and prove Doob’s Optional Sampling Theorem. This result essentially says that no legitimate strategy can alter the nature (that is, from fair to one sided) of a two-person game. The famous identities of Wald follow from this theorem. Doob’s maximal inequality shows how the maximum of a sub-martingale is controlled by the last point of the sequence probabilistically.

6. Chapter 5. Almost Sure and  Convergence

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   This chapter is on convergence of martingales. We state and prove \(L^p\) and almost sure convergence results for martingales, sub-martingales and super-martingales. The crucial tool in developing these results is the Upcrossing lemma. It gives a bound for the expectation of the number of upcrossings of any interval by a sub-martingale, in terms of the expectation of the end point of the sequence and the end points of the interval. There are numerous results on the convergence of martingales, and we provide a sampling of these. The notion of reverse martingale (time-reversed martingales) is also introduced.

7. Chapter 6. Application of Convergence Theorems

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   We present some applications of the results from Chap. 5. We show how Kolmogorov 0-1 law and Hewitt-Savage 0-1 law follow from the reverse martingale convergence theorem. The strong law of large numbers for average of independent and identically distributed (iid) random variables, as well as for U-statistics, are proved by using reverse martingales. The strong law for exchangeable sequences is also established. We also state and prove de-Finetti’s theorem for exchangeable random variables, which says that every such sequence is iid, conditional on an appropriate \(\sigma \)-field. The final topic in this chapter is Kakutani’s theorem for product martingales, which has application in statistics.

8. Chapter 7. Central Limit Theorem

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   This chapter is on the central limit theorem (CLT) for martingales, which is very useful, since it can be used in numerous dependent models. We state and prove one of the simplest versions of CLT for martingales. As simple illustrations, we apply it to a simple urn model, to the trace of a random matrix, and to Markov chains.

9. Chapter 8. Additional Topics

   Arup Bose, Arijit Chakrabarty, Rajat Subhra Hazra

   Abstract

   We cover some additional topics such as forward martingale representation for U-statistics, extended/conditional Borel-Cantelli lemma, Azuma-Hoeffding inequality, conditional three series theorem, strong law for martingales, and the Kesten-Stigum theorem for a simple branching process. The so called Burkholder inequalities are also covered in this chapter.

10. Backmatter