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/(
+ 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,
and
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
/(
+
).