#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

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?


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