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Asymptotic Results for Stochastic Processes via New Projective Methods

通过新投影方法得出随机过程的渐近结果

基本信息

批准号：
2054598
负责人：
Magda Peligrad
金额：
$ 30.22万
依托单位：
University of Cincinnati Main Campus
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054598&HistoricalAwards=false
关键词：
Asymptotic Results Stochastic Processes via

项目摘要

An important area of research in probability theory, with rich applications in statistics, is the asymptotic theory of stochastic processes, also known as large sample theory. This is a framework to assess properties of estimators and statistical tests, necessary for a high level of confidence in predictions. The large sample theory, first developed for independent variables, is considerably more difficult when data is dependent, that is, when quantities have dependencies that relate their values. The questions become even more challenging when the random variations are changing with time. The goal of this research is to address these questions by developing new methods to analyze large samples selected from classes of dependent structures arising in many applied fields, such as data from economics or engineering. The aim of this project is to develop new techniques for studying sequences and fields of dependent random variables, which will lead to sharp inequalities and general limit theorems for both stationary and non-stationary additive functionals of Markov chains and other dependent structures. The PI plans to develop a new type of approximation with martingales for Markov chains and fields based on the fruitful idea of conditioning with respect to both past and future of the process. The advantage of this new, surprising method is that no restrictions on the rate of convergence to zero of the dependence coefficients is required for obtaining various limit theorems. The PI also aims to develop operator perturbation theory for non-stationary Markov chains and functions of independent random elements. This approach will lead to new, deep local limit theorems for Markov chains, random fields, and the left random walk on the group of invertible d-dimensional real matrices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

概率论的一个重要研究领域，在统计学中有着丰富的应用，是随机过程的渐近理论，也被称为大样本理论。这是一个评估估计量和统计检验属性的框架，对于预测的高置信度是必要的。大样本理论最初是为独立变量发展起来的，当数据是相关的时，也就是说，当数量具有与它们的值相关的依赖性时，它就困难得多了。当随机变量随时间变化时，问题变得更具挑战性。本研究的目标是通过开发新的方法来解决这些问题，以分析从许多应用领域中产生的依赖结构类中选择的大样本，如经济学或工程学数据。这个项目的目的是发展新的技术，研究序列和领域的相依随机变量，这将导致尖锐的不等式和一般极限定理的平稳和非平稳添加剂功能的马尔可夫链和其他相关结构。PI计划开发一种新型的近似与鞅的马尔可夫链和领域的基础上，富有成效的想法条件与过去和未来的过程。这个新的，令人惊讶的方法的优点是，没有限制的依赖系数收敛到零的速度是需要获得各种极限定理。PI还旨在发展非平稳马尔可夫链和独立随机元素函数的算子扰动理论。这种方法将导致新的，深刻的局部极限定理马尔可夫链，随机场，和左随机游走组的可逆d维真实的matrix.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。