Explaining a Mysterious Maximal Inequality — and a Path to the Law of Large Numbers

Abstract

In 1964 A. Garsia gave a stunningly brief proof of a useful maximal inequality of E. Hopf. The proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. Here we suggest a straightforward first step analysis that may dispel some of the mystery. The development requires little more than the notion of a random variable, and, the inequality may be introduced as early as one likes in a graduate probability course. The benefit is that one gains access to a proof of the strong law of large numbers that is pleasantly free of technicalities or tricky ideas.

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J. Michael Steele

J. MICHAEL STEELE received his Ph.D. in mathematics from Stanford University. He has taught at U.B.C., Stanford, CMU, Princeton, and the Wharton School of the University of Pennsylvania. He enjoys mathematical inequalities and sequential decision making, and he pays the rent with stochastic calculus and financial time series.