On a closure property of convolution equivalent class of distributions (Q2190030)

The authors of this paper consider the closure problem in the convolution equivalent class of distributions \(\mathcal{S}(\gamma)\), \(\gamma \geq 0\). They show that, if \(F,G \in \mathcal{L}(\gamma)\), then inclusions of \(F*G\), \(FG\), and \(pF+(1-p)G\) for all (some) \(p \in (0,1)\) into the class \(\mathcal{S}(\gamma)\) are equivalent. Also, they provide an example, which shows that \(pF+(1-p)G\) can be in \(\mathcal{S}(\gamma)\) even if \(F,G \notin \mathcal{S}(\gamma)\).