In a text I read, the distribution of a random variable $X$ has so-called regular tails, if the following property holds:
The distribution of $X$ belongs without centering to the domain of attraction of some stable law $\lambda$ with index $\alpha \in (0,2]$. The limit law $\lambda$ is not a one-sided stable law, that is, $0< \lambda (\mathbb{R}^+) < 1$.
I looked up what stable laws and one-sided stable laws are. What I do not understand is the domain of attraction and even more importantly, I don't have any intuition in what sense the tails of the distribution are then 'regular'. Can anyone explain to me why calling it regular makes sense?
In particular, why it is about the regularity of the tails. I do not see the specific connection to the tails.