Random walk with shrinking steps

摘要

We outline the properties of a symmetric random walk in one dimension in which the length of the nth step equals lambda(n), with lambda<1. As the number of steps N&RARR;&INFIN;, the probability that the end point is at x approaches a limiting distribution P-λ(x) that has many beautiful features. For λ<1/2, the support of P-lambda(x) is a Cantor set. For 1/2less than or equal tolambda<1, there is a countably infinite set of λ values for which P-λ(x) is singular, while P-λ(x) is smooth for almost all other λ values. In the most interesting case of λ=g&EQUIV;(&RADIC;5-1)/2, P-g(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)&EQUIV;&INT;(b)(a),P-g(x) dx. (C) 2004 American Association of Physics Teachers.