Convolutions and generalization of logconcavity: Implications and applications

Additive convolution of unimodal and alpha-unimodal random variables are known as an old classic problem which has attracted the attention of many authors in theory and applied fields. Another type of convolution, called multiplicative convolution, is rather younger. In this article, we first focus on this newer concept and obtain several useful results in which the most important ones is that if f circle phi is logconcave then so are F circle phi and (F) over bar circle phi for some suitable increasing functions phi. This result contains phi(x)=x and phi(x)=e(x) as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f(x) and f(e(x)) and two comprehensive implications charts are provided. Then, these fundamental results are applied to aging properties, existence of moments and several kinds of ordered random variables. Multiplicative strong unimodality in the discrete case is also introduced and its properties are investigated. In the second part of the article, some refinements are made for additive convolutions. A remaining open problem is completed and a conjecture concerning convolution of discrete alpha-unimodal distributions is settled. Then, we shall show that an existing result regarding convolution of symmetric discrete unimodal distributions is not correct and an easy alternative proof is presented. (C) 2016 Wiley Periodicals, Inc.