发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对重尾噪声下指数稳定线性系统，基于单条轨迹的最小二乘估计，建立了非渐近样本复杂度界，并推广到亚指数和亚高斯噪声，且证明外生输入模型的误差界维度因子与模型阶数无关。
AI 中文摘要
我们针对指数稳定系统在重尾噪声下基于单条观测轨迹的向量自回归模型的最小二乘估计，建立了非渐近样本复杂度界。通过假设噪声独立同分布、噪声协方差有界以及持续激励条件，我们证明了在p>2且p阶矩有界的情况下，估计误差为$\widetilde{\mathcal{O}}(r^{1/2}T^{-1/2+1/p})$，其中T为样本数，r为噪声维度，$\widetilde{\mathcal{O}}(\cdot)$隐藏了对数项。我们还引入了一种统一的样本复杂度分析方法，适用于广泛的噪声分布类别，并通过推导亚指数和亚高斯噪声分布的误差界来展示其适用性。最后，我们将分析专门应用于带外生输入的向量自回归模型，并证明误差界的维度因子与模型阶数无关。
英文摘要
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.
