A GENERALIZATION OF THE RESULTS OF PILLAI

In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-1 61) showed that the distribution 1 - E(alpha)(-x(alpha)), 0 < alpha le ss-than-or-equal-to 1; 0 less-than-or-equal-to x, where E(alpha)(x) is the Mittag-Leffler function, is infinitely divisible and geometricall y infinitely divisible. He also clarified the relation between this di stribution and a stable distribution. In the present paper, we general ize his results by using Bernstein functions. In statistics, this gene ralization is important, because it gives a new characterization of ge ometrically infinitely divisible distributions with support in [0, inf inity).