On Lévy measures for infinitely divisible natural exponential families

We link the infinitely divisible measure [mu] to its modified Lévy measure [rho]=[rho]([mu]) in terms of their variance functions, where x-2[[rho](dx)-[rho]({0})[delta]0(dx)] is the Lévy measure associated with [mu]. We deduce that, if the variance function of [mu] is a polynomial of degree p[greater-or-equal, slanted]2, then, the variance function of [rho] is still a second degree polynomial. We illustrate these results with some Lévy processes such as positive stable and a class of Poisson stopped-sum processes.