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


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.


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#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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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 $mathbb E_pi[g(X_1)]<infty$. 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!!!