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

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 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. 1. Exploration and First Step Analysis At first, we consider an infinite sequence X,X1, X2, . . . of independent, identically distributed random variables that we assume to have a finite first moment, so in symbols E|X| < ∞. According to custom, we let Sk = X1 +X2 + · · ·+Xk; we think of the index k as time, and we call the sequence {Sk, 1 ≤ k < ∞} a random walk (starting at S0 = 0). We also introduce the maximal process (1) Mn def = max(0, S1, S2, . . . , Sn) 1 ≤ n < ∞, and we emphasize two points: (1) we include zero as a maximand and (2) we do not take absolute values of the partial sums. There are many good, non-mysterious reasons for being interested in Mn, but we leave those reasons aside for the moment. Our first goal is simply to see what one can say about Mn, if we just take one step at a time. The usual aim of such an exploration is to find a pleasing recurrence relation. It would be nice if, after taking our first step, we were to have the identity (2) Mn(ω) = X1 +max(0, X2, X2 +X3, . . . , X2 +X3 + · · ·+Xn), but it is easy to find examples that show that this need not be true. Still, one can ask when it is true, and, if we ponder that possibility for a moment, it may not take long to guess that it is true for all ω such that Mn(ω) > 0. This is a reasonable conjecture, and in two steps one can tease out a confirmation. If Mn(ω) > 0, then random walk has a positive maximum at some time in the interval 1 ≤ k ≤ n. If this maximum occurs at time k = 1, then Mn = X1, the second summand of (2) is zero, and the identity holds. Alternatively, if the strictly positive maximum is attained at some 1 < k ≤ n, then the second summand of (2) is strictly positive. In this case, we can remove the leading zero from the set of maximands, and we see that the identity (2) again holds. J. M. Steele: The Wharton School, Department of Statistics, Huntsman Hall 447, University of Pennsylvania, Philadelphia, PA 19104. Email address: [email protected].