###
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*_{alpha}(-*x*^{alpha}), 0 < *alpha* __<__ 1; 0 __<__
*x*, where *E*_{alpha}(*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.

**Source**
( TeX ,
DVI ,
PS )