On Lévy measures for infinitely divisible natural exponential families

We link the infinitely divisible measure μ to its modified Lévy measure ρ=ρ(μ) in terms of their variance functions, where x-2[ρ(dx)-ρ({0})δ0(dx)] is the Lévy measure associated with μ. We deduce that, if the variance function of μ is a polynomial of degree p⩾2, then, the variance function of ρ 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.