I am seeking papers that construct or prove the existence of joint mass distributions (couplings) of given discrete variables (preferably at least one of which is infinite) with the condition that certain entries in the joint mass table must be 0.

Some entries are 0: In my example, I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. I am placing a restriction on the coupling $(A',B')$ (if such a coupling exists) of $A,B$ by forcing some entries to be 0: $$P(A'=i,B'=j)=0 \text{ when } \frac{i}{\gcd(i,j)} \text{ is composite}.$$.

Recently, (Reference Request for Couplings with Conditions) I learned that Hall's Marriage Theorem may provide a technique to prove the existence of a joint distribution with such contraints.

I would like to see examples (not necessarily using Hall's Theorem) which successfully construct or prove the existence such joint distributions.


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