(Very) short proof of Rayleigh's Theorem (and extensions)

Details (Very) short proof of Rayleigh's Theorem (and extensions) (Very) short proof of Rayleigh's Theorem (and extensions)

2010-07-28

arxiv.org/abs/1007.4870v2

Mathematics

Combinatorics

2 pages

Scientific paper

Consider a walk in the plane made of $n$ steps of length 1, 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 $i$ and $j$ 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~$i/(i+j)$.

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