## 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:

 6e+06 Probability distributions: general theory
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### References:

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