Let $X$ be a non-negative random variable. We call $X$ regularly varying with tail index $\alpha>0$ if $$\lim_{u\to\infty}\frac{\mathbb P[X>ut]}{\mathbb P[X>u]}=t^{-\alpha}, \hspace{1cm}\forall t>0.$$
It is well-known, see e.g. Heavy-Tailed Time Series Proposition 1.4.6., that for such random variables we have $$\forall\beta<\alpha:\mathbb E|X|^\beta<\infty,\hspace{1cm}\forall\gamma>\alpha: \mathbb E|X|^\gamma=\infty,$$ hence we can view $\alpha$ as describing how heavy-tailed the distribution of $X$ is.
Now, more generally for a random variable $X$ define $$\eta_X:=\sup\big\{\beta\geq 0: \mathbb E|X|^\beta<\infty\big\}.$$
From the basic result listed above we have for a regularly varying random variable $X$ with tail index $\alpha$ that $\eta_X=\alpha$. I want to know whether it is possible to derive a result in the other direction:
Are there any known conditions under which a random variable $X$ with $\eta_X\in(0,\infty)$ is regularly varying with index $\eta_X$?
As a first remark, I doubt that $\eta_X$ being finite and positive is sufficient for $X$ to be regularly varying, since by Karamata's Characterization Theorem we can write any regularly varying tail function as $$(*)\hspace{1cm}\mathbb P[X>t]=t^{-\alpha}\cdot \ell(t)$$ for a slowly varying function $\ell$. Now $\eta_X\in(0,\infty)$ only tells us that $$\int_0^\infty \mathbb P[X>t^{1/\beta}]dt<\infty, \hspace{1cm}\forall \beta<\eta_X.$$ However, i doubt that from the finiteness of these integrals alone it is possible to derive a representation as in $(*)$.
