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On Weak Convergence within the ${\cal L}$-Like Classes of Life Distributions

Gwo Dong Lin
Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)
Vol. 60, No. 2 (Jun., 1998), pp. 176-183
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25051196
Page Count: 8
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On Weak Convergence within the ${\cal L}$-Like Classes of Life Distributions
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Abstract

Let ${\cal L}_{\alpha}$, α > 0, be a class of life distributions F with Laplace transform $L_{F}(s)\leq (1+\beta s)^{-\alpha}$ for s ≥ 0, where β = μ(F)/α ≥ 0 and μ(F) stands for the mean of F. Then for each α > 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 $\in \{\frac{1}{m}\}_{m=1}^{\infty}\cup \{-r_{\ast}\}$, where r* ϵ (0, α), 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 ϵ (-α, 1). Based on this new characterization result, a necessary and sufficient condition for weak convergence to a gamma distribution is given.

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