随机序列极值的渐近理论研究

Extreme value theory is concerned with probabilistic and statistical questions related to extreme values in random sequences and in stochastic processes. With rich theoretical support, the subject has been widely used in a variety of areas such as insurance, financial econometrics and ruin theory, where the analysis of asymptotics play an important role. Asymptotic behavior can give many uncovered performance and relationships of risk measures, ruin probabilities, and distributional tails of random summation. The first-order approximations may be the first step to understand of the tail behaviors of risks, ruin probabilities, subordinated distributions. The second step is the convergence rates or second-order approximations which may provide more precise asymptotic information. This project focuses on the study of convergence rates and higher-order asymptotics of extremes of independent and dependent random sequences as the following three parts...Part I focuses on the limit theory of extremes of independent and identically distributed (i.i.d.) random sequences. For i.i.d. random sequences, the convergence rates of distributions and moments of the normalized maxima are studied by de Haan and Resnick (1996), Peng and Nadarajah (2012). Motivated by their work, we investigate the convergence rates of densities of maxima under linear normalization and power normalization, respectively. Since skew distribution families are flexible parametric families with additional parameters allowing to regulate skewness and tails, we consider the univariate random sequences following skew distributions, and establish the uniform convergence rates of distributions of their normalized maxima, the convergence rates and the higher-order expansions of moments of absolute value of their normalized maxima. ..Part II considers the asymptotics of extremes of Hüsler-Reiss model and its extensions. The Hüsler-Reiss model is formed by an independent bivariate Gaussian random triangular array. Hüsler and Reiss (1989) provided that the limiting distribution of maximum is the max-stable Hüsler-Reiss distribution. Recently, the research interest on Hüsler-Reiss distribution has grown significantly mainly due to the fact that not only Gaussian, but elliptical triangular arrays and some more general models have maxima attracted by that distribution, see, e.g., Hashorva (2005, 2006), Hashorva et al. (2012). In this part, we investigate the uniform convergence rate of moment for absolute value of the normalized maxima of Hüsler-Reiss model. By assuming that each vector of the nth row follows from a bivariate Gaussian distribution with correlation coefficient being a monotone, continuous function of i/n, the Hüsler-Reiss model is extended to a non-identically case. For this extended model and other extensions, the higher-order expansions and the convergence rates of extremes are established...Pert III is interested in the copula and the tail asymptotics of the sum of dependent random sequences. Copula is more and more popular since it allows to break away from the standard assumptions which generally underestimate the probability of joint extreme risks. Juri and Wuthrich (2003) derived the convergence results of upper tail dependence copula (UTDC). The results were applied to Archimedean copula, Gaussian copula and elliptical copula, see Juri and Wuthrich (2003), Asimit and Jones (2007). Based on their work, we use second-order regular varying function to study the second-order expansion of UTDC of bivariate elliptical distribution. Since copula can help understand better the various facets of stochastic dependence, the second-order tail asymptotics of the sum of dependent random variables have been analyzed by Kortschak (2012) and Coqueret (2014) under given copulas. In this part, we also consider the second-order tail asymptotics of the sum of dependent heavy-tailed random variables with properties of heavy-tailed distributions and copula.

极值极限理论在金融计量、保险、破产理论等领域已得到广泛应用，其中渐近估计扮演了重要角色，相比于一阶渐近结果，二阶渐近结果对极端事件的预测，及其风险的管理和控制起着更好的指导作用。本课题重在对极值极限理论中收敛速度和高阶展开相关内容进行研究。在已有工作基础上，研究：1)独立同分布随机变量序列极值极限理论，分别在线性赋范和幂赋范情形下，研究随机序列极大值密度函数的收敛速度，及特殊分布极大值分布函数、密度函数、绝对值矩函数的收敛速度和高阶展开；2) Hüsler-Reiss模型及其推广模型的极值极限理论如二维Hüsler-Reiss模型极大值绝对值矩的一致收敛速度，Hüsler-Reiss推广模型极大值分布的收敛速度和高阶渐近展开；3) Copula及具有Copula结构相依随机变量序列极值极限理论特别是二维Copula的高阶渐近展开，及具有Copula结构的相依随机变量部分和尾概率的二阶展开。

极值极限理论在金融计量、保险、破产理论等领域已得到广泛应用，其中渐近估计扮演了重要角色，相比于一阶渐近结果，二阶渐近结果对极端事件的预测，及其风险的管理和控制起着更好的指导作用。本项目主要从三个方面对极值极限理论中收敛速度和高阶渐近展开相关内容进行研究。具体为：..1)一维独立同分布随机变量序列极值极限理论：考虑到有偏分布族能够很好的捕捉到高频数据所呈现出的非对称性，而在金融保险、空间计量、天文学等领域中被广泛应用，项目组主要就服从偏正态分布和有限混合偏t分布的独立同分布随机变量序列极大值、极小值的分布函数和密度函数的高阶渐近行为进行了研究。在线性赋范情形下，得到了服从偏正态分布的独立同分布随机变量序列极大值和极小值联合分布函数和密度函数的高阶渐近展开。在线性赋范和幂赋范情形下，分别建立了服从有限混合偏t分布的独立同分布随机变量序列极大值的分布函数和密度函数的高阶渐近展开式。..2)Hüsler-Reiss模型及其推广模型的极值极限理论: 由于二维独立同分布高斯随机三角阵列极大值两分量之间是渐近独立的，且二元高斯Copula也是尾渐近独立的，这会导致使用二元高斯分布或者二元Copula拟合实际数据时存在误差。因此项目组对极大值两分量渐近相依的二维Hüsler-Reiss模型及其推广模型进行了研究。在幂赋范情形下，得到了二维Hüsler-Reiss模型极大值分布函数的二阶渐近展开。在线性赋范情形下，建立了二维独立椭球随机向量三角阵极大值分布的二阶渐近展开式。..3) Copula及具有Copula结构相依随机变量序列极值极限理论：Copula作为一种刻画随机变量之间相依性的方法，被广泛应用于金融、风险管理等领域。因此项目组主要研究了二维Copula的高阶渐近展开，以及具有Copula结构的相依随机变量部分和尾概率的二阶渐近展开。利用推广的Hüsler-Reiss条件，项目组建立了二维动态高斯Copula极大值分布函数的二阶渐近展开式。对于两类二维偏椭圆分布，得到了它们的上尾相依系数的极限以及收敛速度，并进一步研究了其上尾相依Copula的二阶渐近展开。对于具有Copula结构的相依随机变量序列，我们在尾分布函数为二阶正规变换函数的条件下研究了其部分和尾概率的高阶渐近展开。.