Weighted approximations, self-normalization and long-range dependance

加权近似、自归一化和远程依赖性

• 批准号: 227999-2010
• 负责人: Szyszkowicz, Barbara
• 金额: $ 0.87万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=524163
• 关键词: Weighted; approximations; self; normalization; long

Results and methods of strong and weak approximations have become an integral part of the theory and applications of probability and statistics in the last 40 years. Recent contributions in this area are mainly concerned with weighted approximations of stochastic processes based on i.i.d. observations (cf., e.g., M.Csörgö and L.Horváth, Weighted Approximations in Probability and Statistics, Wiley 1993). We construct new tools for use in weighted approximations of additive processes in different metrics. The long term objective of my research remains to be creating a unified theory of weighted asymptotics of various stochastic processes in different metrics under optimal conditions. This goal has already been achieved for partial sum processes, U-statistics type processes, sums of sequential ranks, empirical processes of observations, ranks and sequential ranks in my earlier papers. Moreover, in [2] and [10], we succeeded in extending Donsker's theorem via weighted sup-norm and Lp-approximations in probability for self-normalized partial sum processes of i.i.d. random variables in the domain of attraction of the normal law (DAN). This in particular extends the theory of invariance principles to partial sums whose variance is not necessarily finite. The construction of these stochastic processes and also that of the earlier ones that I study is such that statistical tests can be based on them for detecting a possible change in their distribution. The problem of abrupt changes arises in areas like quality control, epidemiology, seismic signal processing, or in the study of archaeological sites.

在过去的40年里，强近似和弱近似的结果和方法已经成为概率统计理论和应用的一个组成部分。 最近在这方面的贡献主要涉及加权近似的随机过程的基础上独立同分布。观察（参见，例如，在一个实施例中，M. Csörgö and L. Horváth，Weighted Approximations in Probability and Statistics，Wiley 1993）。我们构建了新的工具，用于加权近似的添加剂过程在不同的度量。我的研究的长期目标仍然是建立一个统一的理论加权渐近的各种随机过程在不同的度量下的最佳条件。这一目标已经实现了部分和过程，U-统计类型的过程，顺序排名的总和，经验过程的观察，排名和顺序排名在我以前的论文。 此外，在[2]和[10]中，我们成功地对独立同分布的自正规化部分和过程，通过加权超模和Lp-概率逼近推广了Donsker定理。在正规律吸引域中的随机变量。这特别地扩展了不变性原理的理论到方差不一定是有限的部分和。这些随机过程的构造以及我所研究的早期随机过程的构造使得统计检验可以基于它们来检测它们的分布中可能的变化。突变问题出现在质量控制、流行病学、地震信号处理或考古遗址研究等领域。