# Large-sample approximations for variance-covariance matrices of high-dimensional time series

## TL;DR

This paper develops large-sample approximations for the variance-covariance matrices of high-dimensional time series, enabling better inference and analysis in complex, high-dimensional data scenarios.

## Contribution

It introduces strong Brownian motion approximations for bilinear forms of sample covariance matrices in high-dimensional time series, covering dependent processes and various statistical applications.

## Key findings

- Valid for high-dimensional, possibly dependent, time series.
- Applicable to sparse principal components and portfolio selection.
- Supports inference in big data contexts.

## Abstract

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $ \ell_1 $-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high--dimensional time series, which matter for the analysis of big data and are discussed in greater detail.