Creating random variables whose pairwise differences are uniformly distributed

Question

I've been banging my head against the wall on this one.

Suppose I have 3 numbers $d_1$, $d_2$, and $d_3$. For example's sake, let's say $(d_1, d_2, d_3) = (2, 3, 4)$ (if I can solve this, then I think I'd know how to generalize). I want to create 3 random variables $r_1, r_2,$ and $r_3$ such that

1. $r_1 - r_2$ is identically the uniform distribution $U(-d_1, d_1)$,
2. $r_2 - r_3$ is identically the uniform distribution $U(-d_2, d_2)$, and
3. $r_3 - r_1$ is identically the uniform distribution $U(-d_3, d_3)$.

If you look at my history you can see a way to do it if say $d_1 = d_2 = d_3$. Solving other cases with $d_i\in\{1, 2, 3\}$ is all readily achievable, but even with a silly case like $(d_1, d_2, d_3) = (2, 3, 4)$ I'm quite stuck.

I'd appreciate any thoughts.

Answer 1

Wew this one took awhile. But here's a way to do it with $d_1 = 2, d_2 = 3$, and $d_3 = 4$:

1. Pick $x$ from $U(-2, 2)$.
2. Set $p_1(x) = x$.
3. Set $p_2(x) = 0$.
4. Toss a coin $c$. If $c$ is Heads, set

$$p_3(x) = \begin{cases} 3x+6 & \text{if }x<-1\\ -3x & \text{if }-1<x<1\\ 3x-6 & \text{if }1<x \end{cases}$$ Otherwise, if $c$ is Tails, set

$$p_3(x) = \begin{cases} -3x-6 & \text{if }x<-1\\ -3x & \text{if }-1<x<1\\ -3x+6 & \text{if }1<x \end{cases}$$

One can verify with conventional methods that all pairwise distributions satisfy the above requirements.

I've included below a graphical view of the above setting. The dashed red lines correspond to the $\frac{1}{2}$ probability of $p_3$ following either one.