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Limit Theorems for Stochastic Processes and Random Fields via Projective Conditions

通过射影条件的随机过程和随机场的极限定理

基本信息

批准号：
1811373
负责人：
Magda Peligrad
金额：
$ 24万
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：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811373&HistoricalAwards=false
关键词：
Limit Theorems Stochastic Processes Random

项目摘要

Many statistical analyses in the social and natural sciences don't take proper account of shifts of distributions. This variability can be seen in the shift in the distribution in stock market, in the income distribution, in the electoral map, in the levels of a river or of the waves. All the data arising from these phenomena are considered to be non-stationary. Non-stationary data, as a rule, cannot be easily modeled and studied. The results obtained by many of the existing models may be spurious. They may indicate a relationship between two variables, where one does not exist. In order to apply consistent, reliable results, the rigorous mathematical tools have to be constructed for non-stationary data. This research has double scope. First, it will contribute to a better understanding and modeling of the non-stationarity phenomena and, in addition, will build new, reliable tools for their forecast and their long term behavior.An important technique for establishing limit theorems for nonstationary sequences and random fields is to approximate them with well understood structures, such as martingales and ortho-martingales. Motivated by the study of asymptotic properties of nonstationary evolutions the theory of nonstationary martingale approximations will be developed. This theory is fundamental for obtaining new projective criteria for nonstationary stochastic processes and for random fields that ensure maximal inequalities and asymptotic results, including the conditional functional central limit theorem and limit theorems started at a point for multi-indexed evolutions. The processes to be studied are very general. They include random fields with long memory and therefore they are useful to model data arising from many applied fields, such as data from economics, social sciences or engineering.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.

社会科学和自然科学中的许多统计分析没有适当考虑分布的变化。这种可变性可以从股票市场的分布变化、收入分配变化、选举地图变化、河流或波浪的水位变化中看出。由这些现象产生的所有数据都被认为是非平稳的。非平稳数据通常不容易建模和研究。现有的许多模型得到的结果可能是虚假的。它们可以表示两个变量之间的关系，其中一个不存在。为了应用一致、可靠的结果，必须为非平稳数据构造严格的数学工具。这项研究具有双重范围。首先，它将有助于更好地理解和建模的非平稳现象，此外，将建立新的，可靠的工具，他们的预测和长期behavior.An建立非平稳序列和随机场的极限定理的一个重要技术是近似它们与理解结构，如鞅和正交鞅。受非平稳演化渐近性质研究的启发，非平稳鞅逼近理论将得到发展。这一理论是获得新的非平稳随机过程和随机场，确保最大的不等式和渐近结果，包括条件功能中心极限定理和极限定理开始于一个点的多指标演化的投影准则的基础。要研究的过程是非常普遍的。它们包括具有长记忆的随机场，因此它们对许多应用领域产生的数据建模非常有用，例如经济学、社会科学或工程学数据。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。