High dimensional random dynamical systems are ubiquitous, including -- but
not limited to -- cyber-physical systems, daily return on different stocks of
S&P 1500 and velocity profile of interacting particle systems around
McKeanVlasov limit. Mathematically, underlying phenomenon can be captured via a
stable $n$-dimensional linear transformation `$A$' and additive randomness.
System identification aims at extracting useful information about underlying
dynamical system, given a length $N$ trajectory from it (corresponds to an $n
\times N$ dimensional data matrix). We use spectral theorem for non-Hermitian
operators to show that spatio-temperal correlations are dictated by the
discrepancy between algebraic and geometric multiplicity of distinct
eigenvalues corresponding to state transition matrix. Small discrepancies imply
that original trajectory essentially comprises of multiple lower dimensional
random dynamical systems living on $A$ invariant subspaces and are
statistically independent of each other. In the process, we provide first
quantitative handle on decay rate of finite powers of state transition matrix
$\|A^{k}\|$ . It is shown that when a stable dynamical system has only one
distinct eigenvalue and discrepancy of $n-1$: $\|A\|$ has a dependence on $n$,
resulting dynamics are spatially inseparable and consequently there exist at
least one row with covariates of typical size $\Theta\big(\sqrt{N-n+1}$
$e^{n}\big)$ i.e., even under stability assumption, covariates can suffer from
curse of dimensionality. In the light of these findings we set the stage for
non-asymptotic error analysis in estimation of state transition matrix $A$ via
least squares regression on observed trajectory by showing that element-wise
error is essentially a variant of well-know Littlewood-Offord problem.

高维随机动力系统中，系统识别的关键是通过观察到的轨迹对系统进行状态转移矩阵估计。由于空间 - 时间相关性，当动力系统具有一个不同的特征值且失配度为 n-1 时，可能导致维度的诅咒问题。通过最小二乘回归处理这些发现来进行误差分析。