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Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function $f(\tau)\ge0$ (that is, $\int f(\tau)d\tau=1$), are there any known results regarding the eigenvalues of the matrix $C_f=\int f(\tau)C(\tau) d\tau$? More specifically, I would like to derive conditions related to stability and thus I am interested in cases of all eigenvalues have negative real part.

I consider here the general case and do not assume detailed balance (that is, $C(\tau)$ is not symmetric). Thank!

EDITED: thinking about it, I'm likewise interested on conditions on $f$ given $C(\tau)$; that is, given such $C(\tau)$, what are the conditions on $f$ so that all eigenvalues have negative real part.

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For a density $f(\tau)$ which is symmetric (around 0) there is a function $\kappa(\tau)$ so that $f(\tau)=\int d\tau'\kappa(\tau')\kappa(\tau'-\tau)$ and in this case have

$$\int f(\tau)C(\tau)d\tau=\int d\tau \int d\tau'\kappa(\tau')\kappa(\tau'-\tau)C(\tau)=\int d\tau \int d\tau'\kappa(\tau')\kappa(\tau)C(\tau'-\tau)=\mathbb{E}[\int d\tau \int d\tau'\kappa(\tau')\kappa(\tau)(s(t-\tau')-\bar s)(s(t-\tau)−\bar s)^T]$$

and thus $\int f(\tau)C(\tau)d\tau=Corr[\kappa*x,\kappa*x]$ which is a possitive-definite and symmetric correlation matrix, with all known results regarding its eigenvalues.

This is a sufficient condition because $\kappa(\tau)$ is defined uniquely by its Fourier transform $\hat{\kappa}(\xi)=\sqrt{\hat{f}(\xi)}$; and it is a necessary condition because requiring $\hat{f}(\xi)=|\hat{\kappa}(\xi)|^{2}$ imply that the Fourier transform of $f(\tau)$ is real and hence it is a symmetric function.

Likewise, for a general density $f(\tau)$ if the correlation is symmetric $C(\tau)=C(-\tau)$ we can divide $f(\tau)$ into a symmetric and anti-symmetric parts $F(\tau)=f_{+}(\tau)+f_{-}(\tau)$ using $f_{\pm}(\tau)=\frac{f(\tau)\pm f(-\tau)}{2}$ and get from the symmetry of $C(\tau)$ and anti-symmetry of $f_{-}$ that $\int f(\tau)C(\tau)d\tau=Corr[\kappa*x,\kappa*x]$ where $\kappa$ is defined by $\hat{\kappa}(\xi)=\sqrt{\hat{f_{+}}(\xi)}$.

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