Consider an exponential dispersion model (EDM) generated by a probability $ ��$ on $[0,\infty )$ which is infinitely divisible with an unbounded L��vy measure $��$. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $��_{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(��_{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(��_0,t)$. Then if $��((x,\infty ))\sim -\ell \log x$ 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 $ ��_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.

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