A Short Proof of Rayleigh's Theorem with Extensions

Abstract

Consider a walk in the plane made of n unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its starting point is 1/(n + 1). We give an elementary proof of this result. We also prove the following generalization, valid for any probability distribution μ on the positive real numbers: If two walkers start at the same point and make, respectively, m and n independent steps with uniformly random directions and with lengths chosen according to μ, then the probability that the first walker ends farther away than the second is m/(m + n).