Consider a random walk starting at the origin in the plane, walking $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, and observing the distance from the origin. Let $b \mapsto W(b;a_1,\ldots,a_n)$ be the cumulative distribution function of this distance.
More formally, if $a_1,\ldots,a_n,b$ are positive real numbers, let $W(b;a_1,\ldots,a_n)$ be the probability that $\|a_1 U_1 + \cdots + a_n U_n\| < b$, where $U_1,\ldots,U_n$ are independent and uniformly distributed on the unit circle.
Can we prove directly that $$ W(a_0;a_1,\ldots,a_n) + W(a_1;a_2,\ldots,a_n,a_0) + \cdots + W(a_n;a_0,\ldots,a_{n-1}) = 1\ ? $$
Can we interpret these $W$'s as the probabilities of $n+1$ disjoint and exhaustive events? Can we give a higher-dimensional generalization?
This relaton is proved in Hermann Weyl's 1938 paper “Mean Motion” (Amer. J. Math. 60 (1938) 889–896), but only incidentally while proving something different (see this other question), and without much commentary.
It may be useful that using the Hankel transform, $W(b;a_1,\ldots,a_n)$ can be expressed as $b \int_0^\infty J_1(bt) J_0(a_1 t) \cdots J_0(a_n t)\,dt$.
