On two broad classes of heavy-tailed distributions

Since the introduction of Class $\mathcal M$ and Class $\mathcal M^*$, they have played important roles in insurance to describe tail equivalence of ruin probability and tail behavior of the deficit at ruin. And in insurance and finance most of heavy-tailed distributions with finite and positive expectations belong to Class $\mathcal M$. So it is important to study tail behaviors of Class $\mathcal M$ and Class $\mathcal M^*$. In this paper, we obtain some results on essential tail behaviors of these two classes.