On \(s\)-convex stochastic extrema for arithmetic risks (Q1962824)

The authors deal with a class of discrete stochastic orderings in the space of random variables. In particular, they follow (their as well as the others authors) former results on this topic. A brief survey of the former results is given in the introduction. After the introduction the authors, first, recall the definition of \(s\)-convex discrete stochastic orderings. The aim of the paper is to construct the extremal (minimal and maximal) distributions with respect to the introduced \(s\)-convex orderings. To this end, first, the general problem of bounding risks is studied. Furthermore, special attention is paid the case when the risks are known to have a decreasing density function. Surely, the investigated problem of stochastic orderings is rather important from the applications point of the view. At the end of the paper, the presented results are applied to obtain the eventual ruin probability in the compound binomial risk process.