The full solution of the convolution closure problem for convolution- equivalent distributions (Q808082)

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.