  We are concerned with the general problem of proving the existence of joint
distributions of two discrete random variables $M$ and $N$ subject to
infinitely many constraints of the form $\mathbb{P}\left(M=i,N=j\right)=0$. In
particular, the variable $M$ has a countably infinite range and the other
variable $N$ is uniformly distributed with finite range. The constraints placed
on the joint distribution will require, for some $j$'s in the range of $N$,
$p\left(i,j\right)=0$ for infinitely many values of $i$ in the range of $M$. To
prove the existence of such a joint distribution, we provide a technique that
furnishes the existence of an $\infty\times n$ matrix consisting of
non-negative real numbers whose row and column sums are known, with zeros in
infinitely many pre-specified locations.
