# #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?

Get this bounty!!!

This site uses Akismet to reduce spam. Learn how your comment data is processed.