Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series A, Pt. 2, pp. 176-183

ON WEAK CONVERGENCE WITHIN THE $\cal L$-LIKE CLASSES OF LIFE DISTRIBUTIONS

By

GWO DONG LIN, Academia Sinica, Taipei

SUMMARY. Let ${\cal L}_\alpha,~ \alpha > 0$, be a class of life distributions $F$ with Laplace transform $ L_F (s) \leq (1 +\beta s)^{-\alpha}$ for $s \geq 0$, where $\beta = \mu(F) / \alpha \geq 0$ and $\mu(F)$ stands for the mean of $F$. Then for each $\alpha>0$, we prove that the ${\cal L}_\alpha$-class of life distributions is closed under weak convergence and that weak convergence is equivalent to the convergence of each moment sequence of order $r\in \{\displaystyle \frac{1}{m}\}_{m=1}^\infty \cup \{-r_*\}$, where $r_*\in(0,\alpha)$, to the corresponding moment of the limiting distribution. This extends Chaudhuri's (1995) result concerning the so-called ${\cal L}$-class (= ${\cal L}_1$-class) of life distributions. We also prove that within the ${\cal L}_\alpha$-class, the gamma distribution is characterized, up to a scale parameter, by one moment of nonzero order $r\in(-\alpha, 1)$. Based on this new characterization result, a necessary and sufficient condition for weak convergence to a gamma distribution is given.

AMS (1991) subject classification. Primary 44A10, 60B10, 62E10.

Key words and phrases. Gamma distribution, characterization of distribution, Laplace transform order, weak convergence.

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