*Bounty: 100*

*Bounty: 100*

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there * necessary and sufficient* conditions

**on $P$**(or on its marginals) ensuring that

$$

exists text{ a random vector $(X_0,X_1,X_2)$ such that }

$$

$$

(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P

$$

Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ **does not imply** that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ **is implied by** $(X_0, X_1, X_2)$ **exchangeable.**

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"

**My thoughts:** among the **necessary** conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that

$$

begin{cases}

P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\

P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\

end{cases}

$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?