Let $X \ge 0$ be a non-negative random variable. I would like to know if the following statements are equivalent:
- $$ \lim_{\lambda \to 0^+} \frac{\mathbb{E} \left[X e^{-\lambda X}\right]}{\log (1/\lambda)} = C.$$
- $$ \mathbb{P}(X > t) = \frac{C}{t} + o(t^{-1}), \qquad t \to \infty.$$
(In the above, the constant $C>0$ is meant to be the same in both statements.)
I suppose this is connected to Tauberian theorems (which I am not familiar with unfortunately) but I am unable to find such form online. It would be great if someone could explain how the equivalence could be proved and even better point me to some useful reference.