#StackBounty: #probability #distributions #random-variable From trivariate cdf to the distribution of differences of random variables

Bounty: 50

Consider a trivariate cumulative distribution function (cdf) $G$.

  • Is there a collection of necessary conditions on $G$ ensuring that
    $$
    exists text{ a random vector $(X_1,X_2)$ such that $(X_1, X_2, X_1-X_2)$ has cdf $G$}
    $$

    ?

  • Is there a collection of necessary and sufficient conditions on $G$ ensuring that
    $$
    exists text{ a random vector $(X_1,X_2)$ such that $(X_1, X_2, X_1-X_2)$ has cdf $G$}
    $$

    ?


Update: Let $P$ be the probability distribution associated with $G$. We can claim that: if there exists a random vector $(X_1,X_2)$ such that $(X_1, X_2, X_1-X_2)$ has probability distribution $P$, then
$$
int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1
$$

  • Is this condition also sufficient? I.e., can we claim that if
    $$
    int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1
    $$

    then
    there exists a random vector $(X_1,X_2)$ such that $(X_1, X_2, X_1-X_2)$ has probability distribution $P$?

  • Can we write
    $$
    int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1
    $$

    by using the cdf $G$
    ?


Get this bounty!!!

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