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


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.


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Stephen J. Wolfe. "On the Behavior of Characteristic Functions on the Real Line." Ann. Probab. 6 (4) 554 - 562, August, 1978.


Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0389.60012
MathSciNet: MR496405
Digital Object Identifier: 10.1214/aop/1176995477

Primary: 60E05
Secondary: 42A72

Keywords: Characteristic function , derivative , smoothness

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • August, 1978
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