Log-concave random variables and their various properties play an increasingly important role in probability, statistics, and other fields. For a distribution F, denote by 𝒟
the set of distributions G such that the convolution of F and G has a log-concave probability mass function or probability density function. In this paper, we investigate sufficient and necessary conditions under which 𝒟
, where F and G belong to a parametric family of distributions. Both discrete and continuous settings are considered.