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

TL;DR

This paper investigates the existence of joint distributions for two discrete variables with infinitely many zero-probability constraints, providing a novel technique to construct such distributions under complex conditions.

Contribution

It introduces a new method to prove the existence of joint distributions with infinitely many zero-probability constraints in discrete variables.

Findings

Established a technique for constructing joint distributions with infinite zero-probability constraints.

Proved the existence of an infinite-by-finite matrix with specified row and column sums and zeros in predetermined locations.

Applied the method to variables with countably infinite and finite ranges.

Abstract

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.