Random walk with shrinking steps

Abstract

We outline the properties of a symmetric random walk in one dimension in which the length of the nth step equals λn, with λ<1. As the number of steps N→∞, the probability that the end point is at x approaches a limiting distribution Pλ(x) that has many beautiful features. For λ<1/2, the support of Pλ(x) is a Cantor set. For 1/2⩽λ<1, there is a countably infinite set of λ values for which Pλ(x) is singular, while Pλ(x) is smooth for almost all other λ values. In the most interesting case of λ=g≡(5−1)/2, Pg(x) is riddled with singularities and is strikingly self-similar. This self-similarity is exploited to derive a simple form for the probability measure M(a,b)≡∫abPg(x) dx.