It is well-known that the cdf (just 'distribution function' from now on) of a discrete random variable is a "staircase function" going from 0 at minus infinity to 1 at plus infinity. Intuitively this is quite clear. I would like to make the notion of 'staircase function' precise, and also to show a "characterization theorem", namely one of the form:
Theorem. Let $X$ be a random variable. Then $X$ is discrete if and only if the distribution function $F_X$ of $X$ satisfies (...).
To further explain I will first recall the basic definitions used here:
Definition. Let $X$ be a random variable. We say that $X$ is finitely discrete if and only if there exists a finite set $C \subseteq \mathbb R$ such that $P( X \in C ) = 1$. More generally we say that $X$ is discrete if and only if there exists a countable set $C \subseteq \mathbb R$ such that $P( X \in C ) = 1$.
For the case when $X$ is what I call 'finitely discrete' (i.e., it can essentially only take on a finite number of values), I think (but I'm not sure) the following holds:
Theorem 1. Let $X$ be a random variable. Then $X$ is finitely discrete if and only if the distribution function $F_X$ of $X$ satisfies
- $\mbox{range} (F_X)$ is finite;
- $F_X$ is non-decreasing;
- $F_X$ is right-continuous at each point;
- $\lim_{x\to -\infty} F_X = 0$ and $\lim_{x\to +\infty} F_X = 1$.
Question. Can someone confirm if this indeed holds?
Now to deal with the more general case, I would like to show the following:
Theorem 2. Let $X$ be a random variable. Then $X$ is discrete if and only if the distribution function $F_X$ of $X$ satisfies
- $\mbox{range} (F_X)$ is countable;
- $F_X$ is non-decreasing;
- $F_X$ is right-continuous at each point;
- $\lim_{x\to -\infty} F_X = 0$ and $\lim_{x\to +\infty} F_X = 1$.
Question. Is the statement from Theorem 2 true? Alas I have not been able to give a proof. If someone could give a hint, or perhaps refer to some literature that details these matters futher, that would be great.
Thanks a lot.