I guess one accepted definition of a ``tail-index" of a random vector $X$ is $\alpha > 0$ (if it exists!) s.t $$P [ \langle u, X \rangle > t] = \Theta (t^{-\alpha}) ~\forall u$$
Can this above definition be referenced to some standard probability theory source?
Now suppose $X$ is s.t for all vectors $u$ and positive integers $k$ we have that $E[ < u, X >^k]$ is finite.
In such a situation it seems that this $X$ cannot have,
(a) inverse polynomially decaying tails
and (b) it cannot have a tail-index defined for it.
- What is the formal theorem which relates the two concepts in these two bullet points and says the above (a) and (b) points?