Limit Theorems for Stochastic Processes and Random Fields via Projective Conditions

社会科学和自然科学中的许多统计分析都没有适当地考虑到分布的变化。这种可变性可以从股市分配的变化、收入分配的变化、选举地图的变化、河流或海浪的水平变化中看出。所有由这些现象产生的数据都被认为是非平稳的。一般来说，非平稳数据不容易建模和研究。现有的许多模型得到的结果可能是虚假的。它们可能表示两个变量之间的关系，其中一个变量不存在。为了应用一致、可靠的结果，必须为非平稳数据构建严格的数学工具。这项研究具有双重视野。首先，它将有助于更好地理解和建模非平稳现象，并将为其预测和长期行为建立新的、可靠的工具。建立非平稳序列和随机场的极限定理的一个重要技术是用人们熟知的结构来逼近它们，如鞅和正序鞅。在研究非平稳演化的渐近性质的基础上，发展了非平稳鞅逼近理论。这一理论是获得非平稳随机过程和随机场的新的射影准则的基础，这些准则确保了极大的不等式和渐近结果，包括条件泛函中心极限定理和多指标演化的极限定理。要研究的过程是非常一般的。它们包括具有长记忆力的随机领域，因此它们对于模拟来自许多应用领域的数据很有用，例如来自经济学、社会科学或工程学的数据。这一奖项反映了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。

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.