Subexponential distribution functions in \(R^{d}\)
In the univariate case a distribution function \(F\), such that \(F(0+)=0\) and \(F(x)<1\) for every \(x\in\mathbb{R}\), is said to belong to the subexponential class \(\mathcal{S}\) if \[ \lim_{x\to +\infty} \overline{F^{*2}}(x)/\overline{F}(x)=2, \] where \(F^{*2}\) denotes the convolution \(F^{*2}=F\ast F\) and \(\overline{F}(x):=1-F(x)\). In the multivariate case, \(F\) belongs to the subexponential class \(\mathcal{S}(\mathbb{R}^d)\) if, and only if, \[ \lim_{t\to +\infty} \overline{F^{*2}}(t\mathbf{x})/\overline{F}(t\mathbf{x})=2, \] whenever \(\mathbf{x}>\mathbf{0}\) and at least one of the components of \(\mathbf{x}\) is finite. Both \(\mathcal{S}\) and \(\mathcal{S}(\mathbb{R}^d)\) contain important special families. The properties of \(\mathcal{S}(\mathbb{R}^d)\) are studied in section 2 (see Theorems 7 and 10, and Propositions 8 and 9). The special case of functions of regular variation is dealt with in section 4.
scientific article; zbMATH DE number 5144820
scientific article