Weighted approximations, self-normalization and long-range dependance

近40年来，强近似和弱近似的结果和方法已成为概率论和统计学理论和应用的重要组成部分。最近在这一领域的贡献主要是关于基于i.i.d观察的随机过程的加权近似（例如M.Csörgö和L.Horváth，概率和统计中的加权近似，Wiley 1993）。我们构建了新的工具，用于不同度量的加性过程的加权近似。我研究的长期目标仍然是在最优条件下创建不同度量下各种随机过程的加权渐近的统一理论。在我之前的论文中，这个目标已经在部分和过程、u统计型过程、顺序秩和、观察的经验过程、秩和顺序秩中实现了。此外，在[2]和[10]中，我们成功地利用加权超范数和概率上的lp逼近，对正态律吸引域（DAN）中i.i.d随机变量的自归一化部分和过程进行了Donsker定理的推广。这特别地将不变性原理的理论扩展到方差不一定有限的部分和。这些随机过程的构造以及我之前研究的那些随机过程的构造是这样的，统计测试可以基于它们来检测它们分布中可能的变化。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。

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.