On discrete α-unimodality

A continuous composition semigroup of probability generating functions F := (F-t, t greater than or equal to 0) and the corresponding multiplication OF of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97-118) are used to introduce the concept of [F; alpha]-unimodality which generalizes the discrete alpha-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;alpha]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;alpha]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F; alpha]-monotone and [F; alpha]-unimodal random variables are derived and some examples are also mentioned.