A note on analytic continuation of characteristic functions

TL;DR

This paper establishes necessary and sufficient conditions for a bounded continuous function to be a characteristic function of a probability measure, using the Cauchy transform and properties of monotonic functions.

Contribution

It introduces a novel analytic framework involving the Cauchy transform to characterize characteristic functions via monotonicity conditions.

Findings

Provides criteria based on the Cauchy transform and its derivatives.

Characterizes characteristic functions using properties of completely and absolutely monotonic functions.

Offers a new perspective on the analytic continuation of characteristic functions.

Abstract

We derive necessary and sufficient conditions for a continuous bounded function $f: R\to C$ to be a characteristic function of a probability measure. The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to the upper and lower half-planes in $C$. The conditions depend on the behavior of $K_f(z)$ and its derivatives on the imaginary axis in $C$. The main results are given in terms of completely monotonic and absolutely monotonic functions.