# A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes

arXiv: https://arxiv.org/abs/2212.09508 · HTML: https://mlsanity.com/paper/2212.09508/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 19 references that are in the map.
- First posted on arXiv 19 Dec 2022.

## Builds on (most cited first, 8 of 8)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Learning without Concentration](https://mlsanity.com/paper/1401.0304/) (2014)
- [Learning with little mixing](https://mlsanity.com/paper/2206.08269/) (2022)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)

## Built on by (most cited first, 1 of 1)

- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)

## Abstract (arXiv)

We present a simple proof for bounding the smallest eigenvalue of the empirical covariance in a causal Gaussian process. Along the way, we establish a one-sided tail inequality for Gaussian quadratic forms using a causal decomposition. Our proof only uses elementary facts about the Gaussian distribution and the union bound. We conclude with an example in which we provide a performance guarantee for least squares identification of a vector autoregression.