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.