On the Dependence of the Component Counting Process of a Discrete Uniform Random Variable

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.