On the Asymptotic Behavior of a Class of Infinitely Divisible Laws

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 3, pp. 330-334

Résumé

The parameters $\gamma,\sigma^2$ and the function $H(u)$ give in P. Levy's formula (1) a certain infinitely divisible law. The class $\mathfrak{H}^\beta$ forms those functions $H(u)$, which for large positive u can be represented in the form $H(u)=u^{-\beta}h(u)$, where $\beta\geq0$ and $h(ku)\sim h(u)$ when $u\to\infty$ for any constant $k > 0$. Theorem 1 proves that for distribution functions of infinitely divisible laws $G(x,\gamma,\sigma^2,H)$, whose function $H$ belongs to one of the $\mathfrak{H}^\beta$, the following asymptotic representation holds true: $$1-G(x,\gamma,\sigma^2,H)\sim-H(x).$$ In Theorem 2 for infinitely divisible laws $\{G_\alpha\}$ of a more restricted one-parameter family $\mathfrak{A}$ than considered in Theorem 1 ($\mathfrak{A}$ is defined in Section 7) the weak and uniform convergence of the functions $G_\alpha(x^{1/\alpha} )$ to the universal law $V(x)$ is proved.