【作者】 刘卫东；

【导师】 林正炎；

【作者基本信息】 浙江大学 ， 概率论与数理统计， 2008， 博士

【摘要】 本文第一部分研究了一类平稳过程(也即X_n=g(…，ε_n-1，ε_n))的各种极限性质，包括部分和的强不变原理，周期图最大值的渐近分布以及谱密度估计的渐近性质．它们是概率统计中十分重要的问题，很多经典的概率统计方面的教科书都对它们有着许多篇幅的介绍．伴随着研究的深入以及当前人们对于非线性时间序列的兴趣，这些问题衍生出了许多新的问题，比如对于非线性时间序列是否也可以考虑类似的问题．鞅逼近是处理平稳过程的一种常用的手段．然而它对于上面提到的问题似乎并不适用，或者说有时候不能达到最理想的结果．鉴于此，本文采用m相依逼近的方法，得到了强不变原理的最优速度，给出了非线性时间序列周期图最大值的渐近分布，也得到了谱密度估计最大偏差的渐近分布．在解决这些问题的同时，我们也解决了近年来一些文献提出的公开的问题．本文第二部分研究了高维向量的独立性检验．高维问题是近年来统计里十分热门的问题．而独立性在许多统计问题里面通常被假设．因而检验独立性是一个十分重要的问题．基于先前文献中的一些工作，我们提出了一个统计量来检验向量问各个分量的独立性，同时也证明了该统计量的渐近分布是极值I型分布，但收敛速度却可以达到多项式速度，比经典的对数收敛速度要快很多．本文第三部分研究了B值空间独立随机变量的重对数律．一直以来，重对数律都是概率极限理论的一个热门的问题．基于最近几年有关这方面的一些文献，我们给出了当方差不存在时的独立不同分布的B值随机变量的重对数律．我们的结果推广了先前人们的一些结论．更多还原

【Abstract】 The first part of this paper concerns some limit properties of a class of stationaryprocesses (i.e. X_n = g(…,ε_n-1,ε_n)), including strong invariance principlesfor partial sums, the maximum of periodograms and asymptotics of the spectral density estimation. These subjects are very important in probability and statistics, and have been explored in many classical textbooks. Because of a surge of interest in nonlinear time series, people proposed many new questions on the subjects referred above. For example, can we consider similar problems for nonlinear time series ? It is well known that martingale approximation is an effective method to deal with stationary processes. However, martingale approximationseems not very suitable for the problems above. For this reason, we use m dependence approximation and obtain the optimal rates for strong invarianceprinciples, the asymptotic distributions of the maxima of periodograms of nonlinear time series and the maximum deviation of the spectral density estimation.Meanwhile, we solve some open questions proposed by some previous papers. The second part of this paper concerns the test on independence between components of a high dimensional vector. The high dimension problem is very popular recently. Since the independence is usually assumed in many statistical problem, test on independence is an important problem. Based on some previous work, we propose a new statistic to test whether components are independent. We also prove that the limit distribution of this statistic is the extreme distributionof type I with a rate of convergence O(（logn）^5/2/（?）). This is much fasterthan O（1/ log n）, a typical convergence rate for this type of extreme distribution. The third part of this paper considers LIL for independent B valued random variables.Based on the previous literature, we prove some LIL for independent B valued random variables when their variances are infinite. Some previous results are extended.更多还原