Autocorrelation function and power spectrum

Define $\tilde C_t=\mathsf{E}\,\xi_t\xi_0^\prime$ which satisfies the ODE $\mathrm{d}\tilde C_t/\mathrm{d}t = A\tilde C_t+a\mu_\infty^\prime$, i.e.

$$\mathsf{E}\xi_t\xi_0^\prime = e^{A\, t}\,V_\infty + \mu_\infty\mu_\infty^\prime \quad .$$ (76)

The auto(cross-)covariance and auto(cross-)correlation functions are therefore

$$\mathsf{cov}_{t-t^\prime} = e^{\vert t-t^\prime\vert\,A}\,V_\infty$$ (77)

$$C_{t-t^\prime} = \mathsf{cov}_{t-t^\prime}\,\mathsf{cov}_0^{-1} = e^{\vert t-t^\prime\vert\,A} \quad .$$ (78)

The Fourier transform of the autocorrelation function is the power spectrum:

$$S(\omega) = \int_{-\infty}^\infty e^{-i\,\omega\,\tau} C_\t... ...ega\,\tau + \vert\tau\vert\,A} = \frac{2 A}{\omega^2 + A^2}$$ (79)

(use $\mathcal{F}(e^{-\gamma \vert s\vert})(\omega) = \frac{2\gamma}{\gamma^2+\omega^2}$.) If the eigenvalues of $A$ are $(\nu_i)_{i=1,\dots,n}$ the eigenvalues of $S(\omega)$ are $( 2\nu_i/(\omega^2+\nu_i^2) )_{i=1,\dots,n}$. If the observed process is an affine function of the state process $\xi_t$, i.e. $y_t = U\xi_t+v$, the observed autocovariance and autocorrelation function are

$$\mathsf{E}\,y_t y_0^\prime - \mathsf{E}y_t\,\mathsf{E}y_0 = U\,C_t\,V_\infty\,U^\prime$$ (80)

$$C^y_t = U\,C_t\,V_\infty\,U^\prime\,\left(U\,V_\infty\,U^\prime\right)^{-1} \quad .$$ (81)

and

$$S^y(\omega) = \int_{-\infty}^\infty e^{-i\,\omega\,\tau} C_... ...\,S(\omega)\,V_\infty\,U^\prime\,(U\,V_\infty\,U^\prime)^{-1}$$ (82)