Mathematical Sciences: Asymptotic Behavior of Dependent Sequences of Random Variables and Applications

本文研究了具有不同类型依赖的随机变量序列的渐近性质。通过将弱相关序列与半鞅和半鞅联系起来，我们将阐明这些序列几乎肯定收敛的一些方面。本研究的另一部分将研究基于相关数据的样本均值方差的两个估计量。这些估计是由阻塞技术和鲁棒统计程序（如bootstrap和jackknife）提出的。这些估计量可以作为中心极限定理中的自归一化量。统计学的主要问题之一是如何对一些未知参数进行区间估计。这些置信区间是基于参数的估计量和由该估计量的可变性给出的误差来构造的。本研究的目的是研究数据依赖情况下的这一问题。

This research involves the investigation of the asymptotic properties of sequences of random variables with various types of dependence. By relating weak dependent sequences to semiamarts and demimartingales we shall clarify some aspects of the almost sure convergence of these sequences. Other part of this research will study two estimators of the variance of the sample mean based on dependent data. These estimators are suggested by blocking techniques and robust statistical procedures such as the bootstrap and the jackknife. The estimators can be used as selfnormalizers in the Central Limit Theorem. One of the main problems of statistics is to provide interval estimates for some unknown parameters. These confidence intervals are constructed based on an estimator of the parameter and an error given by the variability of this estimator. The purpose of this research is to study this problem in the case when the data is dependent.