A GENERALIZATION OF THE RESULTS OF PILLAI

YASUHIRO FUJITA

Department of Mathematics, Toyama University, Toyama 930, Japan

(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−E_{α}(−x^{α}), 0 < α ≤ 1; 0 ≤ x, where E_{α}(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, ∞).

Key words and phrases: Bernstein function, Laplace-Stieltjes transform, infinite divisibility, geometric infinite divisibility, Lévy process.