# Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory

arXiv: https://arxiv.org/abs/2610.00637 · HTML: https://mlsanity.com/paper/2610.00637/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [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)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Outlier-Robust Linear System Identification Under Heavy-tailed Noise](https://mlsanity.com/paper/2501.00421/) (2024)

## Abstract (arXiv)

We establish non-asymptotic sample complexity bounds for the least-squares estimation of vector autoregressive models for exponentially stable systems with heavy-tailed noise based on a single observed trajectory. By assuming i.i.d. noise, bounded noise covariance, and persistent excitation, we show that the estimation error is widetildeO(r1/2T-1/2+1/p) under bounded pth moment for p > 2, where T is the number of samples, r is the noise dimension, and widetildeO(·) hides logarithmic terms. We also introduce a unifying approach to sample complexity analysis applicable to broad classes of noise distributions and showcase this by deriving error bounds for sub-exponential and sub-Gaussian noise distributions. Finally, we specialize our analysis to autoregressive models with exogenous inputs and show that the dimension factor of the error bound is independent of the model order.