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Strictly: No such $B$ exists. You could simply have a sufficient streak of "bad luck" and draw positive $\eta>\epsilon>0$ continuously, for example. Obviously, $B<\lim_{N\to\infty}\sum_{n=-N}^{N-1}\epsilon_n<\lim_{N\to\infty}\left\lvert\sum_{n=-N}^{N-1}\eta_n\right\rvert\,\forall B\in \mathbb R$. Granted, the event that every $\eta_n>...


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BOTTOM LINE UP FRONT: I think the exponential decay growth in $\left<|x(t)|^2\right>$ can be shown in the frequency domain only if the "boundary terms" are nonzero when we compute the Fourier transform of $dx(t)$ from the original SDE. I provide only a start in the work below. Since these processes seem like they could possibly be complex-...


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Given the definition of the correlation matrix $\mathbf{R}_{\mathbf{x}}$ here, I am assuming that $\mathsf{E}[\mathbf{x}] = \mathbf{0}$. I do this because the correlation matrix is usually defined as $\mathsf{E}[(\mathbf{x} - \mathsf{E}[\mathbf{x}])(\mathbf{x} - \mathsf{E}[\mathbf{x}])^{\dagger}]$, where $\dagger$ indicates complex conjugate tranpose. Note ...


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