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.
Citation
Stephen J. Wolfe. "On the Behavior of Characteristic Functions on the Real Line." Ann. Probab. 6 (4) 554 - 562, August, 1978. https://doi.org/10.1214/aop/1176995477
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