Weighted approximations, self-normalization and long-range dependance

在过去的40年里，强逼近和弱逼近的结果和方法已成为概率统计理论和应用的重要组成部分。最近在这一领域的贡献主要涉及基于I.I.D.的随机过程的加权逼近。观察(参看M.Csörgö和L.Horváth，《概率统计中的加权近似》，Wiley 1993)。我们构造了新的工具，用于不同度量下的加性过程的加权逼近。我研究的长期目标仍然是建立一个统一的理论，在最优条件下，在不同的指标下，各种随机过程的加权渐近性。这一目标在我以前的论文中已经在部分和过程、U-统计型过程、序列和、观察的经验过程、序列和以及序列序列中得到了实现。此外，在[2]和[10]中，我们成功地通过加权超范数和Lp-逼近推广了Donsker定理。正态吸引域中的随机变量(DAN)。这特别将不变性原理的理论推广到方差不一定是有限的部分和。这些随机过程以及我研究的早期随机过程的结构是这样的，即统计检验可以基于它们来检测其分布的可能变化。突变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。

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.