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edited May 17, 2021 at 16:37

user44143

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. In other words, startConsider a random walk starting at the origin in the plane, walkwalking $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, thenand observing the distance from the origin. Let $b \mapsto W(b;a_1,\ldots,a_n)$ isbe the cumulative distribution function of the totalthis distance traveled.

(Using the Hankel transform More formally, it turns out thatif $a_1,\ldots,a_n,b$ are positive real numbers, let $W(b;a_1,\ldots,a_n)$ can be expressed asthe probability that $b \int_0^\infty J_1(bt) J_0(a_1 t) \cdots J_0(a_n t)\,dt$, but this should not be relevant to my question$\|a_1 U_1 + \cdots + a_n U_n\| < b$, I only mention it for completenesswhere $U_1,\ldots,U_n$ are independent and uniformly distributed on the unit circle.)

This quantity satisfies the relation: $$ W(a_0;a_1,\ldots,a_n) + W(a_1;a_2,\ldots,a_n,a_0) + \cdots + W(a_n;a_0,\ldots,a_{n-1}) = 1 $$Can we prove directly that This$$ W(a_0;a_1,\ldots,a_n) + W(a_1;a_2,\ldots,a_n,a_0) + \cdots + W(a_n;a_0,\ldots,a_{n-1}) = 1\ ? $$

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 this is doneonly incidentally in the course ofwhile proving something different (see this other question), so Weyl doesn't comment much on this result, and I was hoping it could be done more directlywithout much commentary.

Can we proveIt may be useful that using the above relation directly? IdeallyHankel transform, $W(b;a_1,\ldots,a_n)$ can we interpret itbe expressed as the probabilities of $n+1$ disjoint and exhaustive events? Also, can we give a higher-dimensional generalization?$b \int_0^\infty J_1(bt) J_0(a_1 t) \cdots J_0(a_n t)\,dt$.

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. In other words, start at the origin in the plane, walk $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, then $b \mapsto W(b;a_1,\ldots,a_n)$ is the cumulative distribution function of the total distance traveled.

(Using the Hankel transform, it turns out that $W(b;a_1,\ldots,a_n)$ can be expressed as $b \int_0^\infty J_1(bt) J_0(a_1 t) \cdots J_0(a_n t)\,dt$, but this should not be relevant to my question, I only mention it for completeness.)

This quantity satisfies the relation: $$ W(a_0;a_1,\ldots,a_n) + W(a_1;a_2,\ldots,a_n,a_0) + \cdots + W(a_n;a_0,\ldots,a_{n-1}) = 1 $$ This is proved in Hermann Weyl's 1938 paper “Mean Motion” (Amer. J. Math. 60 (1938) 889–896), but this is done incidentally in the course of proving something different (see this other question), so Weyl doesn't comment much on this result, and I was hoping it could be done more directly.

Can we prove the above relation directly? Ideally, can we interpret it as the probabilities of $n+1$ disjoint and exhaustive events? Also, can we give a higher-dimensional generalization?

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 $b \int_0^\infty J_1(bt) J_0(a_1 t) \cdots J_0(a_n t)\,dt$.

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asked May 13, 2021 at 21:53

Gro-Tsen

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Sum of variables uniformly distributed on a circle: a cyclic property

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. In other words, start at the origin in the plane, walk $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, then $b \mapsto W(b;a_1,\ldots,a_n)$ is the cumulative distribution function of the total distance traveled.

Can we prove the above relation directly? Ideally, can we interpret it as the probabilities of $n+1$ disjoint and exhaustive events? Also, can we give a higher-dimensional generalization?

pr.probability