TITLE: Infinite divisibility of bounded random variables
QUESTION [5 upvotes]: I am trying to prove that if a random variable $X$ is bounded, it cannot be infinitely divisible unless it's degenerate.
My attempt: Suppose not. Let $|X| \leq B$. Since $X$ is infinitely divisible, for any integer $n \in \mathbb{N}$, there exists i.i.d random variables $X_1,\ldots,X_n$ such that $X=\sum_{i=1}^N X_i$. After this step, I think I should prove that each $X_i$ is also bounded by $B/n$ but I don't know how to prove this. Any suggestion would be appreciated.

REPLY [6 votes]: You have the right idea to get started: if $X_1>\frac{B}{n},\dots,X_n>\frac{B}{n}$ then $X>B$, hence
$$ \mathbb{P}\Big(X_1>\frac{B}{n}\Big)^n=\mathbb{P}\left (X_1>\frac{B}{n},\dots,X_n>\frac{B}{n}\right)\leq \mathbb{P}(X>B)=0 $$
and similarly $\mathbb{P}(X_1<-\frac{B}{n})=0$.
Therefore $X_1$ is bounded by $\frac{B}{n}$. Now what does this tell you about $\mathrm{var}\left(X_1\right)$, and therefore about $\mathrm{var}(X)$?