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The full solution of the convolution closure problem for convolution- equivalent distributions. (English) Zbl 0731.60013

A distribution function F belongs to the class S(\(\gamma\)) with \(\gamma\geq 0\) if
(i) \(\lim_{x\to \infty}(1-F^{2*}(x))/(1-F(x))=2\hat f(\gamma)<\infty\),
(ii) \(\lim_{x\to \infty}(1-F(x-y))/(1-F(x))=e^{\gamma y}\forall y\in {\mathbb{R}},\)
where \(F^{2*}\) is the convolution square and \(\hat f\) is the moment generating function of F. Such distribution functions are called convolution-equivalent. We construct an example that proves that none of these classes \({\mathcal S}(\gamma)\) is closed under convolution.

MSC:

60E05 Probability distributions: general theory
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References:

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