AISM 54, 125-137
© 2002 ISM

A notion of \alpha-monotonicity with generalized multiplications

Emad-Eldin A. A. Aly1 and Nadjib Bouzar2

1Department of Statistics and O.R., Kuwait University, P.O.B. 5969, Safat 13060, Kuwait, e-mail:
2Department of Mathematics, University of Indianapolis, Indianapolis, IN 46227, U.S.A., e-mail:

(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.

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