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.