The limiting behavior of some infinitely divisible exponential dispersion models

Consider an exponential dispersion model (EDM) generated by a probability μ on [0,∞) which is infinitely divisible with an unbounded Lévy measure v. The Jørgensen set (i.e., the dispersion parameter space) is then R⁺, in which case the EDM is characterized by two parameters: θ0, the natural parameter of the associated natural exponential family, and the Jørgensen (or dispersion) parameter, t. Denote the corresponding distribution by EDM(θ0,t) and let Y_t be a r.v. with distribution EDM(θ0,t). Then for v((x,∞))~-ℓlogx around zero, we prove that the limiting law F0 of Y_t^-t as t→0 is a Pareto type law (not depending on θ0) with the form F₀(u)=0 for u<1 and the form 1-u^-ℓ for u≥1. This result enables an approximation of the distribution of Y_t to be found for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.