On the Behavior of Characteristic Functions on the Real Line

Abstract

This paper is concerned with the following question: if a characteristic function satisfies a certain property at the origin, what can be said about its behavior on the entire real line? If $k$ is an even integer and $f(u)$ is a characteristic function, then the existence of $f^{(k)}(0)$ implies the existence of $f^{(k)}(u)$ for all $u$. If $k$ is an odd integer, then it is possible to construct a characteristic function $f(u)$ such that $f^{(k)}(0)$ exists but $f^{(k)}(u)$ fails to exist for almost all $u$. However the existence of $f^{(k)}(0)$, when $k$ is odd, implies that $f(u)$ satisfies a $k$th order smoothness condition uniformly on the real line and thus $f(u)$ has many of the properties of a characteristic function with a continuous $k$th derivative. Several other results are obtained that show that if a characteristic function has a property $P$ at 0 then it either has the same property everywhere on the real line or comes close to having the property everywhere.