Asymptotic ordering of distribution functions and convolution semigroups
Let \({\mathcal H}\) be the set of all distribution functions on [0,\(\infty)\), and define \[ {\mathcal L}(\gamma)=\{F\in {\mathcal H}:\quad \lim_{x\to \infty}(1-F(x-y))/(1-F(x))=e^{\gamma y},\quad y\in [0,\infty)\}, \] for \(\gamma\geq 0\). For \(F,G\in {\mathcal H}\), put \(F\leq^ wG\) whenever there is \(c\geq 1\) such that \(1-G(x)\leq c(1-F(x)),\) \(x\in [0,\infty).\) This paper contains a lot of easy results, such as: (a) if \(F_ 1\leq^ wF_ 2\) and \(G_ 1\leq^ wG_ 2\), then \(F_ 1*G_ 1\leq^ wF_ 2*G_ 2;\) (b) if \(F_ 1\leq^ wF_ 2\) and \(G_ 1\leq^ wG_ 2\), then \(pF_ 1+(1-p)G_ 1\leq^ wpF_ 2+(1-p)G_ 2\) for all \(p\in [0,1]\); (c) if \(F\in {\mathcal L}(\gamma)\) and \(G\in {\mathcal L}(\gamma ')\) for \(0\leq \gamma \leq \gamma '\), then \(pF+(1-p)G\in {\mathcal L}(\gamma)\) for all \(p\in (0,1]\).
scientific article; zbMATH DE number 4126388
scientific article