2 - Inequalities and Laws of Large Numbers

Abstract

This chapter describes inequalities and laws of large numbers. Almost all proofs of martingale limit theorems rely to some extent on inequalities. This chapter presents the application of inequalities to obtain laws of large numbers. There is a very strong case for including the martingale convergence theorem as a strong law of large numbers, especially as the classical law for sums of independent and identically distributed r.v. is a corollary. If ▪ is a submartingale and ϕ is a nondecreasing convex function on ▪ such that ▪ for each n, then ▪ is also a submartingale. The requirement that ϕ be nondecreasing may be dropped if ▪ is a martingale. The square function inequalities form a major contemporary addition to the armory of martingale tools. They imply a rather unexpected relationship between the behavior of a martingale and the squares of its differences. This duality was noticed earlier for sums of independent r.v. and orthogonal functions, and many of the inequalities are generalizations of this work.