Game-Theoretic Foundations for Probability and Finance

3Some Basic Supermartingales

In Section 1.4, we discussed two simple supermartingales. One was quadratic in Reality's moves; the other was exponential. This chapter studies some additional quadratic and exponential supermartingales, which appear in various guises in various protocols. They can be used directly and in the construction of more complex supermartingales. We will see some of them when we study the law of the iterated logarithm (Section 5.4), defensive forecasting (Lemma 12.10), and continuous‐time probability (in the construction of the Itô integral in Chapter 14 and the construction of portfolios in Chapter 17).

Quadratic supermartingales can grow moderately fast and are often used to show that moderately large deviations from expectations (i.e. from prices given by Forecaster) are unlikely. Exponential supermartingales can grow much faster. They are sometimes competitive with quadratic supermartingales for moderately large deviations, and they can be used to show that very large deviations are very unlikely.

In Section 3.1, we study a classic quadratic martingale, which we call Kolmogorov's martingale, after Kolmogorov's early work on the law of large numbers for dependent outcomes 220,221. Then we turn to some exponential supermartingales, to which we attach the names of Catherine Doléans (Section 3.2), Wassily Hoeffding (Section 3.3), and Sergei Bernstein (Section 3.4). Doléans's supermartingale was implicit in our proof of Borel's law of large numbers in Chapter ...