Soft: Applications of Entire Functions to Probability Theory

Question

What are some applications of entire functions to probability theory?

For example, I know many random variables' characteristic function can be extended to a certain strip of the complex plane, but I've never seen any extension theorems to all of $\mathbb{C}$ itself.

Answer 1

An example that comes to mind is Cramer's decomposition theorem:

If $X$ and $Y$ are independent so that $X + Y$ is a normal random variable, then both $X$ and $Y$ are normal.

The standard proof is by entire functions. See these notes for a proof, and more references to the use of entire functions.