On Representation of the Logarithm for Arbitrary Characteristic Function on Segments

We consider a characteristic function of an arbitrary probability law. We obtain analogs of the Lévy–Khintchine formula for it on any segment of the form [−r, r] with finite r > 0, where the characteristic function does not vanish. Using these representations, we prove a criterion of belonging of the corresponding distribution function to a new wide class of so-called quasi-infinitely divisible distribution functions.