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.