The Dominant Relations Of Heavy-tailed Distributions And The Precise Asymptotics For The Point Process Of The Jump Times Of An Extremal Process In Insurance And Finance

In order to meet the needs of recent research in Applied Probability, such as finance and insurance, risk theory, random walk theory, queueing theory and branching processes and so on, the concepts of heavy-tailed random variables (or heavy-tailed distributions) are introduced. They are one of the important objects many scholars are concerned on. On the other hand, in a risk process, the number of these heavy-tailed variables' occurrence until the time t, i.e. all kinds of counting process, is one of the important objects, which many scholars are studying. This paper mainly discusses the dominant relations of heavy-tailed distributions, and the precise asymptotics for the point process of the jump times of an extremal process.Since there are too many subclasses of heavy-tailed distributions and their relations are very complicated, in Chap 2 of this paper, we'll discuss the dominant relations of heavy-tailed distributions, and obtain the equivalent condition of the distributions which can dominate all light-tailed distributions, as well as those which can dominate all light-tailed distributions and lightly heavy-tailed distributions, in Section 2.2. In Section 2.3, we'll discuss the dominant relations between distributions and their corresponding equilibrium distributions. In the last section of this chapter Section 2.4, we'll apply the dominant relations in the closure of sums; In Chap 3, we'll discuss the precise asymptotics. We'll introduce some definitions and some previous results in Section 3.1, and introduce the precise asymptotics for the counting process of record times of i.i.d. r.v.s in Section 3.2, all of which extend and generalize the corresponding results in Gut (2002) [19]. At last, in Section 3.3, we'll emphasize on the precise asymptotics for the point process of the jump times of an extremal process.