Discrete ss-convex extremal distributions: Theory and applications

Given a nondegenerate moment space with ss fixed moments, explicit formulas for the discrete ss-convex extremal distribution have been derived for s=1,2,3s=1,2,3 (see [M. Denuit, Cl. Lefèvre, Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences, Insurance Math. Econom. 20 (1997) 197–214]). If s=4s=4, only the maximal distribution is known (see [M. Denuit, Cl. Lefèvre, M. Mesfioui, On ss-convex stochastic extrema for arithmetic risks, Insurance Math. Econom. 25 (1999) 143–155]). This work goes beyond this limitation and proposes a method for deriving explicit expressions for general nonnegative integer ss. In particular, we derive explicitly the discrete 4-convex minimal distribution. For illustration, we show how this theory allows one to bound the probability of extinction in a Galton–Watson branching process. The results are also applied to derive bounds for the probability of ruin in the compound binomial and Poisson insurance risk models.