Discrete [formula omitted]-convex extremal distributions: Theory and applications

Abstract

Given a nondegenerate moment space with s fixed moments, explicit formulas for the discrete s -convex extremal distribution have been derived for s = 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 = 4 , only the maximal distribution is known (see [M. Denuit, Cl. Lefèvre, M. Mesfioui, On s -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 s . 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.