In my book about risk, there is a chapter where they show that \begin{align*} \mathbb{P}(\max\{X_1, X_2\} > x) = \mathbb{P}(X_1 > x) + \mathbb{P}(X_2 > x) - \mathbb{P}(X_1 > x, X_2 > x) = 2 \overline{F}(x) - \overline{F}^2(x) \sim 2 \overline{F}(x) \end{align*} where $X_1, X_2$ are independent and $\overline{F}(x)$ denotes the tail function for a distribution function $F$ on $(0,\infty)$.. From this, they conclude that the condition that $F$ is subexponential, i.e. that $\overline{F}^{*n}(x)/\overline{F}(x) \rightarrow n$ as $x \rightarrow \infty$, is that the probability of the set $\{X_1 + X_2 > x\}$ is asymptotically the same as the probability of its subset $\max\{X_1, X_2\} > x$. Therefore, in the subexponential case, the only way that $X_1 + X_2$ can get large is of either $X_1$ or $X_2$ is becoming large.
Can anyone try to explain how they conclude this from the fact that $2 \overline{F}(x) - \overline{F}^2(x) \sim 2 \overline{F}(x)$? I do not seem to understand why.
TIA for any help.
$$\frac{P(X_1+X_2 > x)}{P(\max\{X_1,X_2\} > x)}=\frac{\overline{F^{*2}}(x)}{2\overline{F}(x)}\frac{2\overline{F}(x)}{ 2 \overline{F}(x) - \overline{F}^2(x) } \to 1$$
as $x \to \infty$. This means $P(X_1+X_2 > x) \sim P(\max\{X_1,X_2\} > x)$ as $x \to \infty$.