I've been trying to prove the next characterization and looked in several books. But all of their proofs seem a little poor. In fact, many of them say that is easy and let the proof as an exercise. But I can't figure a nice way to prove it yet:
Let $X>0$ be a random variable with probability distribution function $F(x)$ and $\overline{F}(x):=1-F(x)$. We define the "mean residual hazard function" as:
$$ \mu_F(x):=\mathbb{E}(X-x|X>x)=\int_x^\infty\frac{\overline{F}(y)}{\overline{F}(x)}dy $$
Proof that if $\displaystyle\lim_{x\to\infty}\mu_F (x)=\infty$, then $X$ is heavy-tailed.
Remember, $X$ is heavy tailed if its moment generating function $\left(\hat{m}_F (s)\right)$ satisfy:
$$ \hat{m}_F (s) = \infty \hspace{1cm} \forall s>0. $$
The statement comes from the book: "Stochastic Processes For Insurance And Finance", page 68, Tomasz Rolski
Does anyone has any idea of how to prove it?