The limiting behavior of some infinitely divisible exponential dispersion models

Consider an exponential dispersion model (EDM) generated by a probability $ \mu $ on $[0,\infty )$ which is infinitely divisible with an unbounded L\'{e}vy measure $\nu $. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $\theta _{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(\theta _{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(\theta_0,t)$. Then if $\nu ((x,\infty ))\sim -\ell \log x$ around zero we prove that the limiting law $F_0$ of $ Y_{t}^{-t}$ as $t\rightarrow 0$ is of a Pareto type (not depending on $ \theta_0$) with the form $F_0(u)=0$ for $u<1$ and $1-u^{-\ell }$ for $ u\geq 1$. Such a result enables an approximation of the distribution of $ Y_{t}$ for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.