In this work, we study non-asymptotic bounds on correlation between two time realizations of stable linear systems with isotropic Gaussian noise. Consequently, via sampling from a sub-trajectory and using \emph{Talagrands'} inequality, we show that empirical averages of reward concentrate around steady state (dynamical system mixes to when closed loop system is stable under linear feedback policy ) reward , with high-probability. As opposed to common belief of larger the spectral radius stronger the correlation between samples, \emph{large discrepancy between algebraic and geometric multiplicity of system eigenvalues leads to large invariant subspaces related to system-transition matrix}; once the system enters the large invariant subspace it will travel away from origin for a while before coming close to a unit ball centered at origin where an isotropic Gaussian noise can with high probability allow it to escape the current invariant subspace it resides in, leading to \emph{bottlenecks} between different invariant subspaces that span $\mathbb{R}^{n}$, to be precise : system initiated in a large invariant subspace will be stuck there for a long-time: log-linear in dimension of the invariant subspace and inversely to log of inverse of magnitude of the eigenvalue. In the problem of Ordinary Least Squares estimate of system transition matrix via a single trajectory, this phenomenon is even more evident if spectrum of transition matrix associated to large invariant subspace is explosive and small invariant subspaces correspond to stable eigenvalues. Our analysis provide first interpretable and geometric explanation into intricacies of learning and concentration for random dynamical systems on continuous, high dimensional state space; exposing us to surprises in high dimensions

翻译：本文研究了各向同性高斯噪声下稳定线性系统两个时间实现之间的非渐近相关性。通过子轨迹采样并利用Talagrand不等式，我们证明奖励的经验均值以高概率集中于稳态（当闭环系统在线性反馈策略下稳定时，动态系统混合到的状态）奖励。与通常认为谱半径越大样本相关性越强的观点相反，系统特征值的代数重数与几何重数之间的巨大差异会导致与系统转移矩阵相关的大规模不变子空间；一旦系统进入大规模不变子空间，它将在远离原点运动一段时间后，才接近以原点为中心的单位球，此时各向同性高斯噪声能够以高概率使其逃离当前所在的不变子空间，从而在张成$\mathbb{R}^{n}$的不同不变子空间之间产生瓶颈。准确而言：起始于大规模不变子空间的系统将长时间滞留其中，滞留时间与不变子空间维数成对数线性关系，与特征值模量倒数的对数成反比。在通过单条轨迹进行系统转移矩阵的普通最小二乘估计问题中，若与大规模不变子空间相关的转移矩阵谱具有爆炸性，而小规模不变子空间对应稳定特征值，则该现象更为显著。我们的分析首次为连续高维状态空间上随机动态系统的学习与集中性问题提供了可解释的几何视角，揭示了高维空间中的意外现象。