【作者】 杨洋；

【导师】 王岳宝；

【作者基本信息】 苏州大学 ， 概率论与数理统计， 2003， 硕士

【摘要】 在应用概率的许多领域，如金融保险、风险理论、随机游动理论、排队论、分支过程等，重尾随机变量或重尾分布都是重要的对象之一，另一方面，在一个风险过程中，到t时刻时，这些重尾变量出现的个数，即各种记数过程，也是人们研究的主要对象之一，本文主要对重尾分布的控制关系与极值过程的跳时点过程的精致渐近性进行深入的讨论。 由于重尾分布族分类众多，各子族之间的关系复杂，本论文将在第二章讨论重尾分布间的控制关系，在2．2节得到了能控制一切轻尾分布的等价条件，以及能控制一切轻度重尾分布，从而也能控制一切轻尾分布的分布的等价条件，2．3节主要讨论了分布F与其相应的均衡分布F_e之间的控制与归属关系，本章最后一节2．4节给出了上述控制关系在和的分布的封闭性等方面的初步应用；在第三章我们将对精致渐近性问题进行研究。我们将在3．1节介绍一些定义及已有的结果，在3．2节介绍记录时记数过程的精致渐近性，这些结果一般化了Gut(2002)[19]中的定理，3．3节给出极值过程的跳时点过程的精致渐近性的定理及推论。更多还原

【Abstract】 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.更多还原