5 - Conditioning and Martingales

Summary

Because they are not needed earlier, conditional expectations do not appear until Chapter 5. The advantage gained by this postponement is that, by the time I introduce them, I have an ample supply of examples to which conditioning can be applied; the disadvantage is that, with considerable justice, many probabilists feel that one is not doing probability theory until one is conditioning. Be that as it may, Kolmogorov’s definition is given in §5.1 and is shown to extend naturally to both σ-finite measure spaces and random variables with values in a Banach space. Section 5.2 presents Doob’s basic theory of real-valued, discrete parameter martingales: Doob’s Inequality, his Stopping Time Theorem, and his Martingale Convergence Theorem. In the last part of §5.2, I introduce reversed martingales and apply them to DeFinetti’s theory of exchangeable random variables.