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Spectral Statistics of the Sample Covariance Matrix for High Dimensional Linear Gaussians
Performance of ordinary least squares(OLS) method for the \emph{estimation of high dimensional stable state transition matrix} $A$(i. e., spectral radius $\rho(A)<1$) from a single noisy observed trajectory of the linear time invariant(LTI)\footnote{Linear Gaussian (LG) in Markov chain literature} system $X_{-}:(x_0, x_1, \ldots, x_{N-1})$ satisfying \begin{equation} x_{t+1}=Ax_{t}+w_{t}, \hspace{10pt} \text{ where } w_{t} \thicksim N(0, I_{n}), \end{equation} heavily rely on negative moments of the sample covariance matrix: $(X_{-}X_{-}^{*})=\sum_{i=0}^{N-1}x_{i}x_{i}^{*}$ and singular values of $EX_{-}^{*}$, where $E$ is a rectangular Gaussian ensemble $E=[w_0, \ldots, w_{N-1}]$.
From Spectral Theorem to Statistical Independence with Application to System Identification
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.
Learning and Concentration for High Dimensional Linear Gaussians: an Invariant Subspace Approach
In this work, we study non-asymptotic bounds on correlation between two time realizations of stable linear systems with isotropic Gaussian noise.
Concentration Phenomenon for Random Dynamical Systems: An Operator Theoretic Approach
Via operator theoretic methods, we formalize the concentration phenomenon for a given observable `$r$' of a discrete time Markov chain with `$\mu_{\pi}$' as invariant ergodic measure, possibly having support on an unbounded state space.
Transportation-Inequalities, Lyapunov Stability and Sampling for Dynamical Systems on Continuous State Space
We study the concentration phenomenon for discrete-time random dynamical systems with an unbounded state space.
Learning Expected Reward for Switched Linear Control Systems: A Non-Asymptotic View
In this work, we show existence of invariant ergodic measure for switched linear dynamical systems (SLDSs) under a norm-stability assumption of system dynamics in some unbounded subset of $\mathbb{R}^{n}$.
