Variance upper bounds and convolutions of \(\alpha\)-unimodal distributions (Q1341358)

The authors revisit the field of discrete \(\alpha\)-unimodal distributions, analogues to distributions of random variables of the form \(YU^{1/\alpha}\), where \(U\) is uniform on (0,1) and independent of \(Y\). These distributions were first considered by the first author [Stat. Neerl. 41, No. 4, 239-244 (1987; Zbl 0691.60009)]. Section 2 provides definitions, Section 3 upper bounds and other inequalities for variances, and Section 4 results about the \(\alpha\)-unimodality of convolutions of \(\alpha\)-unimodal distributions. Similar results by the same authors have appeared in a recent paper [Stat. Papers 35, 151-161 (1989)].