Some results of characteristic functions and characterizations of the normal and generalized stable laws

Abstract

A variety of result are presented in this paper. Some results are established concerning how behaviour of a characteristic function (e.f.) in the neighbourhood of the origin affects the existence of moments of the corresponding distribution function (d.f.); it is shown, inter alia, that if two c.f.'s coincide at a sequence of points tending to the origin and one of them corresponds to a d.f. having moments of all orders and uniquely determined by its moments, then the two c.f.'s coincide. These results, which find application in the later sections, are presented in Section 2. We introduce and study a class of distributions which we call "generalized stable (GS) laws", in Section 3. These provide a natural generalization of the stable laws and include the class semi-stable laws considered by P.Levy. Our main results concern characterizations (through regression) of the Normal and the GS laws; these are stated in Section 1 and proved in Section 4.