(Received April 20, 1992; revised August 3, 1992)
Abstract. In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-161) showed that the distribution 1 - Ealpha(-xalpha), 0 < alpha < 1; 0 < x, where Ealpha(x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in [0,\infty).
Key words and phrases: Bernstein function, Laplace-Stieltjes transform, infinite divisibility, geometric infinite divisibility, Lévy process.