Central Limit Theorems and Inference in High Dimensions

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.