AISM 54, 125-137

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

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.