AISM 54, 125-137

© 2002 ISM

## A notion of \alpha-monotonicity with generalized multiplications

### Emad-Eldin A. A. Aly^{1} and Nadjib Bouzar^{2}

^{1}Department of Statistics and O.R., Kuwait University,
P.O.B. 5969, Safat 13060, Kuwait, e-mail: emad@kuc01.kuniv.edu.kw

^{2}Department of Mathematics, University of Indianapolis, Indianapolis, IN 46227, U.S.A., e-mail: nbouzar@uindy.edu

(Received February 1, 2000; revised August 24, 2000)

Abstract.
The multiplications of van Harn et al. (1982, *Z. Wahrsch. Verw. Gebiete*, **61**, 97-118) are used to generalize the definition of $\alpha$-monotonicity of Olshen and Savage (1970, *J.
Appl. Probab.*, **7**, 21-34) and Steutel (1988, *Statist. Neerlandica*, **42**, 137-140) for distributions with support in $\pmb{$Z$}_+$ and $\pmb{$R$}_+$. Several characterizations are offered and a convolution property is established. Some relevant stability equations are
solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the $\pmb{$R$}_+$-case from those for the $\pmb{$Z$}_+$-case.

Key words and phrases:
Semigroup, monotonicity, self-decomposability, Poisson mixtures.

**Source**
(TeX , DVI )