## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!

## #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!!!