Self-normalized processes, U-statistics, changepoint analysis, long-memory processes, anisotropic random walks

提案标题：自归一化过程、U-统计量、变点分析、长记忆过程、各向异性随机游动 我们首先引用伯特兰·罗素的一句话：“尽管这似乎是一个悖论，但所有的科学都被近似的思想所支配。”不变性原理涉及理论上可识别的各种可观测(基于数据)过程的近似率。因此，前者在原则上和实践上都可以由后者来研究。在这样做的同时，还必须意识到底层流程中可能发生的变化。事实上，突变的问题出现在各种实验和数学科学中。例如，在流行病学中，人们可能对检验一种疾病的发病率是否随时间保持不变而感兴趣，如果不是，则对估计变化的时间(S)(S)以提出可能的原因感兴趣。另一个例子是心电图中的节律分析，其中变化检测方法的使用构成了模式识别分析的一部分。检测地震信号过程中的突变是我们通常所说的变点分析的另一个例子。自归一化(部分)和过程的不变性原理理论最近已成为这方面信息和灵感的主要来源之一。当用一个过程来近似另一个过程时，还应该考虑这样做时可能出现的错误。这进而导致研究由此产生的独立序列和长范围相关序列的误差过程的波动规律。所有这一切提供了统计、概率和随机过程理论和应用的各个领域的方法的美丽相互作用。特别是，研究高斯和相关随机过程的路径性质的需要进入并加强了我们建议中提到的所有领域的研究，包括物理文献中也研究的各向异性随机游动的那些领域。

Title of Proposal: Self-normalized processes, U-statistics, Changepoint Analysis, Long-memory processes, Anisotropic random walks We start with a quotation from Bertrand Russel: "Although this may seem a paradox, all science is dominated by the idea of approximation." Invariance Principles are concerned with rates of approximation of various observable (data based) processes by theoretically recognizable ones. Thus the former can, in principle and also in practice, be studied by the latter. While doing this one must also be aware of possible changes in the underlying processes. Indeed, the problem of abrupt changes arises in a variety of experimental and mathematical sciences. For instance, in epidemiology one may be interested in testing whether the incidence of a disease has remained constant over time, and if not in estimating the time(s) of change(s) in order to suggest possible causes. Another example is rhythm analysis in electrocardiograms, where the use of change detection methods constitutes a part of pattern recognition analysis. Detecting abrupt changes in seismic signal processes is yet another example of what we, in general, call Changepoint Analysis. The theory of invariance principles for Self-normalized (partial) Sums processes has recently become one of the main sources of information and inspiration in this regard. When approximating a process by another one, one should also think of the possible errors made while doing this. This, in turn, leads to studying the laws of fluctuation of the thus created Error Processes for independent, and also for long-range dependent sequences. All this provides a beautiful interplay of the methods of various fields of the theory and applications of Statistics, Probability and Stochastic Processes. In particular, the need for studying path properties of Gaussian and related stochastic processes enters and strengthens the research done in all the areas mentioned in our proposal, including those of anisotropic random walks that are also studied in the physics literature.