Characteristic functions of a class of probability distributions

Abstract

The characteristic function of a class of continuous one-sided probability distributions is being considered. The distribution class contains three independent parameters, one of them represents scale, the other two determine initial and terminal shape of the associated probability density function. The analytical properties of the characteristic function depend heavily on the terminal shape parameter λ which may vary in the interval (-∞, 1). If 0 >λ>1, the characteristic function is many-valued with branch points at zero and infinity. Its principal branch is holomorphie and bounded upon analytic continuation (into the complex plane cut along the nonnegative real axis) from the primary element which is holomorphie in the open left-hand plane. If λ = 0. the primary element of the characteristic function is holomorphie in the half-plane left of the vertical line through the point (b-1, 0)b being the scale parameter. Upon continuation it becomes either a rational function (if the initial shape parameter is a nonpositive integer) with a pole at the point (b-1, 0) or a many-valued function with branch points at (b-1, 0) and infinity whose principal branch is holomorphie in the plane cut along the real axis from b-1 to infinity. If λ>0, the characteristic function is an entire function of order greater than unity. It has no real zeros but an infinity of conjugate complex pairs of zeros even if the order is an even integer.