I have a stable linear dynamical system $x_t \in \mathbb{R}^d, 1\leq t \leq T$ such that $$ x_t = A x_{t-1}+N_t, \quad N_t \sim \mathcal{N}(0,I_d), x_0=0. $$ Define $\hat{\Gamma}_t=\frac{1}{n}\sum_i x_t^{(i)}(x_t^{(i)})^\top$ and ${\Gamma}_t=\mathbb{E}[x_t x_t^\top]$ where $\{x_t^{(i)} :1 \leq i \leq n, 1 \leq t \leq T \}$ denotes the collection of $n$ i.i.d trajectories generated according to the system.

I am wondering if we can provide concentration inequalities for the $\hat{\Gamma}_t$ to center around $\Gamma_t$. In particular, the bounds of the type

$$ \mathbb{P}\left(\| (\hat{\Gamma}_1-\Gamma_1)+(\hat{\Gamma}_2-\Gamma_2)+\ldots+(\hat{\Gamma}_T-\Gamma_T) \|_{\mathrm{op}} \geq \varepsilon \right) \leq e^{-f(d, n,\varepsilon,T)}, $$ for some function $f$ depending on $d, n,\varepsilon$ and $T$.

Can anyone provide useful links or resources regarding where do people use such concentration bounds for dynamical systems?


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