On discrete α-unimodality

A continuous composition semigroup of probability generating functionsF≔(F _t ,t≥0) and the corresponding multiplication ⊙ _F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of F;α]-monotone and F;α]-unimodal random variables are derived and some examples are also mentioned. Research supported by Grant SS024 of the Research Center of Kuwait University.