*Bounty: 50*

*Bounty: 50*

Let $mathcal{P}$ be the family of continuous distribution functions in $mathbb{R}^3$ whose marginals are **symmetric around zero** and **identical**.

Fix a vector of reals $thetaequiv (theta_1, theta_2)in mathbb{R}^2$. Fix a vector of positive reals $pequiv (p_1,p_2,p_3)in mathbb{R}^3_{+}$ such that $p_1+p_2+p_3=1$.

**Assumption 1:** There exists $Pin mathcal{P}$ such that

$$

begin{aligned}

& p_1=PBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)\

& p_2=PBig((-infty,infty)times [-theta_2,infty)times (-infty, theta_2-theta_1)Big)\

& p_3=PBig((-infty, -theta_1)times (-infty,-theta_2)times (-infty,infty)Big)\

end{aligned}

$$

**Clarification about the notation**: $PBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)$ denotes the probability measure of the box $[-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)$. That is,

$$

PBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)=PBig({(x,y,z): (x,y,z)in [-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)}Big)

$$

**Note:** I’m not restricting the support of $P$. However, note that $P$ cannot have full support on $mathbb{R}^3$ under Assumption 1. For instance, Assumption 1 implies that

$$

P({(x,y,z): xgeq -theta_1, y<-theta_2, z<theta_2-theta_1})=0

$$

**Question:** I would like your help to show **Claim 1**:

**Claim 1**: If Assumption 1 holds, then there exists a distribution $Hin mathcal{P}$ whose support is

$$

mathcal{B}equiv {(b_1, b_2, b_3)in mathbb{R}^3: b_1=b_2+b_3}

$$

and such that

$$

begin{aligned}

& p_1=HBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)\

& p_2=HBig((-infty,infty)times [-theta_2,infty)times (-infty, theta_2-theta_1)Big)\

& p_3=HBig((-infty, -theta_1)times (-infty,-theta_2)times (-infty,infty)Big)\

end{aligned}

$$

**Clarification about the notation**: $HBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)$ denotes the probability measure of the box $[-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)$. That is,

$$

HBig([-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)Big)=HBig({(x,y,z): (x,y,z)in [-theta_1,infty)times (-infty,infty)times [theta_2-theta_1,infty)}Big)

$$

**Sub-question**: What is the role played by the requirement that $mathcal{P}$ contains distributions whose marginals are symmetric around zero and identical (if any)? That is, if I characterise $mathcal{P}$ as being another family of distributions (for instance, distributions with zero first moment), does the answer change?

**This may be helpful:** Take a distribution $H$ whose support is

$$

mathcal{B}equiv {(b_1, b_2, b_3)in mathbb{R}^3: b_1=b_2+b_3}

$$

I believe that this is equivalent to impose

$$

H(mathcal{A}(b_1, b_2))=H(mathcal{C}(b_1, b_2))=0

$$

for each $(b_1, b_2)in mathbb{R}^2$, where

$$

begin{aligned}

&mathcal{A}(b_1, b_2)equiv {(x,y,z)in mathbb{R}^3: x>b_1+b_2, yleq b_1, zleq b_2}\

&mathcal{C}(b_1, b_2)equiv {(x,y,z)in mathbb{R}^3: xleq b_1+b_2, y> b_1, z> b_2}\

end{aligned}

$$

I’ve done several simulations which confirm Claim 1. Clearly, such simulations are not a formal proof. In case you have a counterexample of Claim 1 in mind, please advise.