Convolutions and generalization of logconcavity: Implications and applications
研究了单峰和对数凹随机变量的加性卷积与乘性卷积，得到乘性卷积保持对数凹性的条件，并应用于老化性质、矩存在性和有序随机变量，同时修正了离散单峰分布卷积的一个错误结论。
Abstract Additive convolution of unimodal and α ‐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 is logconcave then so are and for some suitable increasing functions ϕ. This result contains and as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f ( x ) and 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 α ‐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. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 109–123, 2016