Gumbel分布最大吸引场的次序统计量的精致渐近性

得到一类Gumbel分布最大吸引场的随机容量样本的次序统计量的精致渐近性,揭示了收敛速度、权函数、边界函数及极限状态之间的联系.这类吸引场真包含了全体(γ),γ>0分布族.     

U-统计量的精致渐近性

设{X_n．n≥1}是一非退化的i．i．d．随机变量序列,U_n是以二维Borel可测对称函数h(x,y)为核函数的U-统计量．记U_n=2/(n(n-1))Σ_≤i≤j≤nh(X_i,X_j)．本文分别在核函数h(x,y)只有4/3阶矩或4/3+δ,0＜δ≤1的情况下,对非常广泛的一类权函数(x)与边界函数b(x)得到了如下关于U-统计量U_n的精致渐近性:不仅使得已有的结果成为我们的特况,还大大降低了其中的矩条件．     

独立与NA列部分和的精致渐近性

令{X,Xi:i≥1)为属于非退化稳定分布一般吸引场的独立同分布随机变量列,Sn=∑i=1nXi.本文对十分广泛的拟权函数和边界函数得到了Sn的精致渐近的结果和正态吸引场强平稳NA列部分和的精致渐近的结果,改进并推广了Wang等的结果,从而使此前关于独立列和NA列部分和精致渐近性方面的许多经典的和最新的结果都可以包括在本文的框架之内.     

极值过程的跳时点过程的精致渐近性

以统一的形式,得到了极值过程的跳时点过程的精致渐近性的一般结果,揭示了精致渐近性研究中拟权函数,边界函数,收敛速度以及极限状态之间的有机关系,从而可以给出许多新的具体结果。     

一类渐近统计量的显式表示

设(X_i)_(i≥1)是一列正的独立同分布随机变量,服从指数为α(0α1)的Pareto型分布.定义■,且记τ_k:=lim_(n→∞)E(T_n~k).利用组合学方法,渐近矩τ_k用Bell多项式予以显式表示,并且得到了用对数型Bell多项式的简明表示,简化和完善了文献中的现有结果.     

Hill统计量的渐近正态性

的正整数。此后,统计量γ_n的渐近性质被广泛地加以研究。Mason证明了γ_n是弱相容的,同时指出:如果k_n=[n~β],0<β<1,那末γ_n也是γ的强相容估计。[3]和[4]讨论了γ_n的渐近正态性;作为正则变化函数性质的一个应用,[5]中对此亦有讨论。本文将在d.f.F连续这一假定下,揭露γ_n的分布与一个条件iid的rv阵列行和的     

关于记录值序列部分和的渐近正态性

证明了Burr分布和Frechet分布的记录值序列的部分和是渐进对数正态的，从而解决了[2]中的一个悬而未决的问题。     

一类向量高斯过程之上穿过点过程的渐近分布

｛Ｘ（ｔ），ｔ≥｝为ｐ维高斯过程，在一定条件下，本文得到了｛Ｘ（ｔ），０≤ｔ≤Ｔ｝对水平ＵＴ（＞０）的ε－上穿过次数所形成的点过程的渐近分布（Ｔ→∞）。证明了Ｐ个分量点过程的渐近独立性。     

一类差分方程的渐近性

本文对高维非自治差分方程x(τ+1)=f(τ,x)+g(τ,y);y(τ+1)=k(τ,x,y) (1)得到了保证其平凡解渐近稳定,一致渐近稳定的判别准则,而对V逐数的差分不要求负定性,推广了文[1]中相应的结果.     

独立非同分布二元高斯三角阵最大值的渐近性及相关统计推断

考虑二元独立非同分布高斯随机向量三角阵列最大值分布的渐近性及相关统计推断.此高斯三角阵的第n列的第i个向量服从二元高斯分布,其相关系数为i/n的函数并单调连续.首先建立了此高斯三角阵最大值分布的一阶和二阶渐近展开式.其次,分析相关系数参数估计及估计量的渐近性质.最后,通过随机模拟说明了相关系数之参数估计的有效性,并将该二元非同分布三角阵列模型应用于实际数据,得到了满意的结果.     

Precise asymptotics for a type of order statistics

This paper obtains some results on the precise asymptotics for the order statistics generated by the random samples of maximum domain of attraction of the Fréchet distribution, which reveal the relations among the boundary function, weight function, convergence rate and limit position in a uniform form.Research supported by National Science Foundation of China (NO. 10271087).     

A General Law of Precise Asymptotics for the Complete Moment Convergence

The authors achieve a general law of precise asymptotics for a new kind of complete moment convergence of i.i.d, random variables, which includes complete convergence as a special case. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. This extends and generalizes the corresponding results of Liu and Lin in 2006.     

自正则Chung型重对数律的精确渐近性

设X,X_1,X_2,…为零均值、非退化、吸引域为正态吸引场的独立同分布随机变量序列,记S_n=■X_j,M_n=■|S_k|,V_n~2=■X_j~2,n≥1.证明了当b>-1时,■δ~(-2(b 1))■(log log n)~P/(n log n)P(Mn/V_n≤ε~(π~2)/(8lgo log n)~(1/2)) =4/πГ(b 1)■~(-1)~k/(2k 1)~(2b 3).     

On Subexponential Distributions and Asymptotics of the Distribution of the Maximum of Sequential Sums

We study the properties of subexponential distributions and find new sufficient and necessary conditions for membership in the class of these distributions. We establish a connection between the classes of subexponential and semiexponential distributions and give conditions for preservation of the asymptotics of subexponential distributions for functions of distributions. We address similar problems for the class of the so-called locally subexponential distributions. As an application of these results, we refine the asymptotics of the distribution of the supremum of sequential sums of random variables with negative drift, in particular, local theorems.     

Strong Domain of Attraction of Extreme Generalized Order Statistics

It is a well-known result in extreme value theory that the von Mises conditions imply the strong convergence of extreme order statistics. We extend this result to extreme generalized order statistics. A characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given. In particular, we determine the asymptotic joint distribution of (upper and lower) extreme generalized order statistics. Finally, we show that the Hill estimator based on extreme generalized order statistics is asymptotic normal.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

NA序列的重对数矩收敛的精确渐近性

假设{X_n,n≥1}为一列严平稳的NA随机变量,期望为零,方差有限.设S_n=∑_(i=1)~n∑X_i,M_n=max_(1≤i≤n)|S_i|.在适当的条件下,得到了一类NA序列部分和部分和最大值重对数矩收敛的精确渐近性.