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: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=478671
• 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.

近40年来，强近似和弱近似的结果和方法已成为概率统计理论和应用的一个组成部分。 该领域最近的贡献主要涉及基于 i.i.d 的随机过程的加权近似。观察（例如，参见 M.Csörgö 和 L.Horváth，概率与统计中的加权近似，Wiley 1993）。我们构建了新的工具，用于不同指标的加性过程的加权近似。我研究的长期目标仍然是创建一个统一的最优条件下不同度量下的各种随机过程的加权渐近理论。在我早期的论文中，部分求和过程、U 统计类型过程、顺序排序总和、观察的经验过程、排序和顺序排序已经实现了这一目标。 此外，在[2]和[10]中，我们通过独立同分布的自归一化部分和过程的概率中的加权超范数和Lp近似成功地扩展了Donsker定理。正常法则（DAN）吸引力域中的随机变量。这尤其将不变性原理理论扩展到方差不一定有限的部分和。这些随机过程以及我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其分布的可能变化。突然变化的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。