Abstract
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 $\widetilde{\mathcal{O}}(r^{1/2}T^{-1/2+1/p})$ under bounded $p$th moment for $p > 2$, where $T$ is the number of samples, $r$ is the noise dimension, and $\widetilde{\mathcal{O}}(\cdot)$ 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.
Keywords
Subject
Publication details
- Journal
- Not available
- Open access
- Green open access
Cite this article
APA 7
Yang, X., & Shin, S. (2026). Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory. https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory
MLA 9
Yang, Xiaomian, and Sungho Shin. "Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory." https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory.
Chicago (author–date)
Yang, Xiaomian, and Sungho Shin. 2026. "Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory." https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory.
Harvard
Yang, X. and Shin, S. (2026) 'Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory', Available at: https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory.
Vancouver
Yang X, Shin S. Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory. https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory
IEEE
X. Yang, and S. Shin, "Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory," https://omanscience.com/en/articles/learning-linear-systems-under-heavy-tailed-noise-a-non-asymptotic-analysis-from-a-single-trajectory.