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We are concerned with the general problem of proving the existence of jointdistributions of two discrete random variables $M$ and $N$ subject toinfinitely many constraints of the form $\mathbb{P}\left(M=i,N=j\right)=0$. Inparticular, the variable $M$ has a countably infinite range and the othervariable $N$ is uniformly distributed with finite range. The constraints placedon 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 $M$. Toprove the existence of such a joint distribution, we provide a technique thatfurnishes the existence of an $\infty\times n$ matrix consisting ofnon-negative real numbers whose row and column sums are known, with zeros ininfinitely many pre-specified locations.

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