On sufficient conditions for subexponentiality (Q1048544)

A distribution function \(F\) on \([0,\infty)\) is said to be subexponential if \[ 1-(F*F)(x)\sim2(1-F(x)),\quad \text{as}\;x\to\infty. \] The author gives conditions ensuring subexponentiality in cases where, as \(x\to\infty\), either \(1-F(x)\) decreases more rapidly than \(e^{-x^{\alpha}}\) for \(\alpha\) arbitrarily close to 1, or \[ 1-F(x)=e^{-g(x)+o(1)}, \] with \(g(x)\) continuous and piecewise differentiable such that \(xg'(x)<\alpha g(x)\) for \(x>x_0\) and some \(\alpha<1\).