# Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series

arXiv: https://arxiv.org/abs/2604.16219 · HTML: https://mlsanity.com/paper/2604.16219/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2026.

## Built on by (most cited first, 3 of 3)

- [High-dimensional Gaussian Graphical Model Testing for Long-Memory Time Series](https://mlsanity.com/paper/2609.30565/) (2026)
- [Valid and Efficient Split Conformal Regression for Time Series](https://mlsanity.com/paper/2609.33866/) (2026)
- [Conformal Coverage of Time Series: Validity and Inference](https://mlsanity.com/paper/2609.33868/) (2026)

## Abstract (arXiv)

For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.