Let $T$ be a real valued absolutely continous r.v. such that $\lim\limits_{t\to +\infty} t^a \mathbb{P}(T>t)=1$ for some $a<1$, is it true that its expected value must be infinite?
I found the related statement in some notes I am reading, and being a bit rusty on these things I might really use some help.
It is straightforward to see this if $T$ is non-negative, but is it true also if $T$ possibly takes on negative values?If that is not the case, which would be a counterexample?
Many thanks in advance!
