Central Limit Theorems and Inference in High Dimensions

中心极限定理(CLT)是概率论中的一个基本结果，它断言由小的和近似独立的随机变量组成的大集合的聚集行为遵循普遍规律，非正式地称为钟形曲线。CLT近似是统计推断的基石，因为它为估计、假设检验和可信区间的绝大多数统计方法提供了理论基础。它还经常用于科学领域和许多工业应用中的不确定度评估。尽管CLT非常受欢迎，但大多数现有的CLT结果无法在数学上完全表达现代大型数据集的典型复杂程度。此外，它们不适用于高维统计建模。因此，依赖经典的CLT近似来验证统计程序的有效性在高维环境中不再合理。相反，新的、更精致的CLT担保是必要的。最近的突破性进展导致了新的高维CLT(HDCLT)的提出，其有效性仍然有效，但只有在相当严格的假设下才成立。这个项目的主要目标是发展新的HDCLT近似，适用于更广泛的条件和环境，并阐明它们在涉及大数据和复杂数据的高维统计问题中的应用。本项目的研究内容包括五个主要研究目标：(1)产生允许重尾数据和奇异协方差的弱条件下的独立观测的HDCLT；(2)推导新的高维Edgeworth展开式；(3)获得相关随机向量和时间序列过程的新的HDCLT；(4)研究高维随机矩阵(如样本协方差矩阵)的HDCLT及其谱；(5)将HDCLT应用于以FDR和FWER控制为目标的多重测试问题。该项目的研究成果将在HDCLT近似的理论和应用方面取得重要进展，并将导致在高维统计中进行推理的新的、实用的工具。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。

The Central Limit Theorem (CLT) is a fundamental result in probability theory, asserting that the aggregate behavior of large ensembles of small and approximately independent stochastic quantities follows a universal law, informally known as the bell curve. The CLT approximation is a cornerstone of statistical inference, as it provides the theoretical underpinning of the vast majority of statistical methods for estimation, hypothesis testing, and confidence intervals. It is also routinely used for uncertainty assessment across the sciences and in many industrial applications. Despite the immense popularity, most existing CLT results are unable to fully express mathematically the degree of complexity that is typical of modern, large datasets. In addition, they are inadequate for high-dimensional statistical modeling. As a result, reliance on classic CLT approximations to verify the validity of statistical procedures is no longer justifiable in high-dimensional settings. Instead, new, more refined CLT guarantees are in order. Recent breakthrough advances have led to the formulation of new high-dimensional CLTs (HDCLTs), whose validity holds but only under fairly restrictive assumptions. The broad goal of this project is to develop new HDCLT approximations that are applicable across a significantly wider range of conditions and settings and to elucidate their uses in high-dimensional statistical problems involving large and complex data.The research components of this project include five main research aims: (1) to produce HDCLTs for independent observations under weak conditions that allow for heavy-tail data and singular covariances; (2) to derive new high-dimensional Edgeworth expansions; (3) to obtain new HDCLTs for sums of dependent random vectors and time series processes; (4) to study HDCLTs for high-dimensional random matrices, such as the sample covariance matrix, and their spectra; and (5) to deploy HDCLTs in multiple-testing problems targeting FDR and FWER control. The research outcomes of this project will advance in important ways the theory and applications of HDCLT approximations and will lead to novel, practicable tools for inference in high-dimensional statistics.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.