The moment index of minima (II)

The moment index κ(X)=sup{k:E(Xk)<∞} of a nonnegative random variable X has the property that κ(min(X,Y))⩾κ(X)+κ(Y) for independent r.v.s X and Y. We characterize conditions under which equality holds for a given r.v. X and every independent nonnegative r.v. Y, and discuss extensions to related r.v.s and their distributions.