Sufficient conditions for the subexponential property of the convolution of two distributions

Conditions on the distributions of two independent nonnegative random variables $X$ and $Y$ are given for the sum $X+Y$ to have a subexponential distribution, i.e., $(1-F^{(2*)}(t))/(1-F(t))\to2$ as $t\to+\infty$, where $F(t)=\mathsf P\{X+Y\le t\}$ and $F^{(2*)}(t)$ is the convolution of $F(t)$ with itself.