## #StackBounty: #probability #distributions #cumulative-distribution-function Show that the support restriction is not a binding constrai…

### 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

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.

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!

## #StackBounty: #probability #distributions #expected-value #calculus When will \$mathbb{E}[g(S_n/n)]\$ exist given \$mathbb{E}[g(X_1)]\$ e…

### Bounty: 50

Suppose $$X_1, X_2,…, X_n$$ are i.i.d. random variables with distribution $$pi$$ on some probability space. Let $$g$$ be a measurable function such that $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$mathbb E_pi[g(S_n/n)]$$, where $$S_n = sum_{k=1}^n X_k$$?

My guess is the quantity $$mathbb E_pi[g(S_n/n)]$$ is not necessarily finite in general, but should be finite if $$g$$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

Get this bounty!!!