On Symmetric Moments and Local Behavior of the Characteristic Function

Abstract

Let $F(x)$ be a probability distribution function and $\phi(t)$ its characteristic function. Define $\psi(t) = \Im \phi(t) = - \int^\infty_0 \sin tx dG(x)$, where $G(x) = 1 - F(x) - F(-x)$ is the tail difference of $F$. In general $G$ is known only to be of bounded variation on $\lbrack 0, \infty\rbrack$. The purpose of this paper is to relate differentiability of $\psi$ at 0 and Lipschitz behavior of $\psi$ to asymptotic behavior of $G$. Let $C_\alpha$ be the class of real valued functions $g$ such that $\int^\infty_x|dg(u)| = o(x^{-\alpha})(x\rightarrow + \infty)$, and let $\mu_\alpha^\ast$ be the symmetric moment of order $\alpha$. Then it is shown that $G \in C_1$ implies $\psi'(0) = \mu_1^\ast$ if either exists, while $G \in C_\alpha$ implies $\psi \in \operatorname{Lip} \alpha$ for $0 < \alpha < 1$. Extensions of these results to higher derivatives and values of $\alpha > 1$ are indicated. Finally it is shown that, for $1 < \alpha < 3, G(x) = o(x^{-\alpha}) (x \rightarrow + \infty)$ and $t^{-\alpha-1}(\psi(t) - t\mu_1^\ast) \in L(0,1)$ imply that $\mu_alpha^\ast$ exists.