(Very) short proof of Rayleigh's Theorem (and 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 $\mu$ 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 $\mu$, then the probability that the first walker ends farther than the second is $m/(m+n)$.