Asymptotic Results for Stochastic Processes via New Projective Methods

概率论的一个重要研究领域是随机过程的渐近理论，也称为大样本理论，在统计学中有着广泛的应用。这是一个评估估计者和统计检验的性质的框架，对于预测的高度置信度是必要的。大样本理论最初是为自变量而发展的，当数据是相依的时，也就是说，当数量具有与其值相关的相依关系时，大样本理论要困难得多。当随机变化随时间变化时，问题变得更加具有挑战性。这项研究的目的是通过开发新的方法来分析从许多应用领域中出现的相关结构类别中选择的大样本来解决这些问题，例如来自经济学或工程学的数据。这个项目的目的是发展研究相依随机变量序列和场的新方法，这将导致马尔可夫链和其他相依结构的平稳和非平稳可加泛函的尖锐的不等式和一般的极限定理。PI计划开发一种新的基于马尔可夫链和马氏场的基于过去和未来条件条件的富有成效的想法的鞅逼近。这种新的、令人惊讶的方法的优点是，不需要限制相关系数收敛到零的速度来获得各种极限定理。PI还旨在发展非平稳马氏链和独立随机元函数的算子摄动理论。这一方法将导致关于马尔可夫链、随机场和可逆d维实矩阵群上的左随机游动的新的、深刻的局部极限定理。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.