On analytic continuation of characteristic functions of probability measures

We derive necessary and sufficient conditions for a continuous bounded function f :R -> 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. (C) 2012 Elsevier Inc. All rights reserved.