The limiting behavior of some infinitely divisible exponential dispersion models

Consider an exponential dispersion model (EDM) generated by a probability μμ on [0,∞)[0,∞) which is infinitely divisible with an unbounded Lévy measure νν. The Jørgensen set (i.e., the dispersion parameter space) is then R+R+, in which case the EDM is characterized by two parameters: θ0θ0, the natural parameter of the associated natural exponential family, and the Jørgensen (or dispersion) parameter, tt. Denote the corresponding distribution by EDM(θ0,t) and let YtYt be a r.v. with distribution EDM(θ0,t). Then for ν((x,∞))∼−ℓlogxν((x,∞))∼−ℓlogx around zero, we prove that the limiting law F0F0 of Yt−t as t→0t→0 is a Pareto type law (not depending on θ0θ0) with the form F0(u)=0F0(u)=0 for u<1u<1 and the form 1−u−ℓ1−u−ℓ for u≥1u≥1. This result enables an approximation of the distribution of YtYt to be found for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.