Modified Iteration and Probability

Abstract

It is always a pleasant event when apparently disparate areas of mathematics intersect: when a theorem in one discipline translates into, proves, generalizes, simplifies or explains a theorem in another. This phenomenon occurs frequently between probability theory and various aspects of classical analysis. Examples are so abundant that it is perilous even to begin a list. This note presents another such interaction. Iteration of functions produces fixed points, at least under suitable conditions. We are going to present a modification of the usual iteration procedure which has the pernicious property of retarding or even destroying the convergence to the fixed point and ascertain (purely analytically) just when this convergence is destroyed. Our modification comes about not purely by malice, but turns out to correspond to picking a non-optimal strategy in a certain iterative game of chance. The analytic criterion then reappears as an immediate corollary to an elementary pointwise convergence theorem. Here is the analytic set-up. G(w) is a strictly convex, continuously differentiable function on the interval 0 6 w 0, G(1) = 1, and 0 < G'(w) < 1 for 0 < w < 1. These conditions are more than enough to ensure that if mqnl1 = G(qn) (io = 0), then qn -* 1, the fixed point of G. Let wn be a sequence with 0 < wn < 1. Modify the in to get a sequence (n by defining 40 = 0 and