Randomization/virtual-re-sampling methods, Changepoint detection, Short and long memory processes, Self-normalized partial sums processes, Planar random walks, Strong and weak approximations

该提案由四个不同的研究计划组成，这些计划基于申请人及其合作者在各自出版物中取得的最新进展。提案(1)涉及探索我们感兴趣的观察样本中所包含的信息的性质，例如流行病学中某种疾病的发病率。我们考虑了在可用数据或大或小的情况下进行推理的问题。对于样本相对较小的情况，我们提出了我们最近建立的随机化技术的多种扩展，与相同条件下的经典过程相比，该技术产生的推理误差显著较小。我们的随机化技术将为众所周知的基于重复重新抽样的计算密集型随机化技术提供直接有效的替代途径，这是我们要避免的。当数据集太大而无法处理时，或者当它构成来自超级总体的随机样本时，我们将使用我们的随机化技术来基于更小的子样本进行推理。我们相信，本研究的预期结果对理论和应用统计学家都很重要，也将有利于各种研究领域中数据处理的理论和实践。建议(2)是探索直接应用我们最近的成果的可能性，即通过随机样本的观测值和的自（随机）归一化来估计观测值的理论平均值（平均），当它可以存在时。这将通过用随机实体（其性质原则上是可计算的）来表示我们最近结果的理论结果，从而使感兴趣的模拟变得可行来实现。我们还将探讨将提案(1)的随机化技术应用于手头环境的可能性。这项工作的预期意义与提案(1)相似。我们关于蜘蛛图上的随机漫步和布朗运动的建议(3)，蜘蛛图是有多条腿的蜘蛛的“图片”，将有助于数学科学的基础知识，最终可能对研究不同科学领域的复杂系统和互联网流量模型有用。已经在进行中的提案(4)的研究受到了一些重要论文的启发，这些论文在一定程度上是由所谓的统计物理的传输现象所激发的，也就是说，所有统计性质的不可逆过程都源于分子的随机连续运动，主要是在流体中观察到的。我们希望对在模拟输运现象中占主导地位的两种各向异性随机游走方法进行相当全面的比较分析。**

This PROPOSAL consists of four distinct research plans that are based on recent advances made in each, in the publications of the applicant with his collaborators. PROPOSAL (1) is concerned with exploring the nature of information contained in a sample of observations that are of interest to us, like for example incidence of a disease in epidemiology. We consider the problem of making inference when the available data is either small or big. For the case of having a relatively small sample, we propose a manifold extension of our recently established randomization technique that yields a significantly smaller magnitude of error of inference as compared to that of the classical procedures under the same conditions. Our randomization technique will provide a direct efficient alternative route to a well known computationally intensive randomization technique that is based on repeated re-sampling, that we are to avoid. When the data set is too big to be processed, or when it constitutes a random sample from a super-population, we are to use our randomization technique to base inference on significantly smaller sub-samples. We believe that the anticipated outcomes of this research will be important to theoretical and applied statisticians alike, and will also benefit the theory and practice of data processing for inference in various research fields of interest. PROPOSAL (2) is to explore the possibility of direct applications of our recent achievements in estimating the theoretical mean (average) of observations when it can exist at all, via self (random)- normalization of sums of observations of a random sample. This will be done by making simulations of interest feasible by representing the theoretical outcomes of our recent results in terms of random entities whose properties are, in principle, computable. We'll also explore the possibility of adapting the randomization techniques of PROPOSAL (1) to the context in hand. The anticipated significance of this work is similar to that of PROPOSAL (1). Our PROPOSAL (3) on random walks and Brownian motion on a spider graph, a 'picture' of a spider with a number of legs, will contribute to basic knowledge in mathematical sciences that may eventually become useful in studying complex systems in different areas of science and internet traffic models. The already ongoing studies of PROPOSAL (4) have been inspired by important papers that, in part, were motivated by the so-called transport phenomena of statistical physics, i.e., all irreversible processes of statistical nature stemming from the random continuous motion of molecules, mostly observed in fluids. We hope to have a fairly comprehensive comparative analysis of two approaches to anisotropic random walks that have been dominant in modeling transport phenomena. **