A Class of Infinitely Divisible Mixtures

Abstract

I X/(X h(t)) are considered. It will be shown that mixtures of cf's of type I are inf div if h(t) is such that X/(X h(t)) is a cf for all X > 0. The class of functions h(t) satisfying this condition will be determined. 2. Preliminaries. In our proof we will make use of the L6vy-Khinchine canonical representation: +(t) is an inf div cf if and only if (2) log +(t) = ait + f {eitx1e -itx/(l + x2)}(1 + x2)x2d0(x), where a is a real constant and 0(x) is bounded and non-decreasing (see e.g. [2], p. 89). Further we shall need the well-known fact (cf. [2], p. 203) that a function of the type