Cross-spectrum of two stationary time series ($f_{XY}(\lambda )=\frac{1}{2\pi }{\sum }_h\mathbb{E}[X_t{Y}_{t+h}]e^{-ih\lambda }$)

For two real centered time series $X_t,Y_t$, require joint weak stationarity: all auto- and cross-covariances depend only on the time lag. Individual weak stationarity alone does not impose this condition on their joint behavior. With the convention ${\gamma }_{XY}(h)=\mathbb{E}[X_t{Y}_{t+h}]$, define the cross-spectrum byAssume absolute summability of the auto- and cross-covariance sequences, so the displayed Fourier series and the two individual spectral densities are continuous. Reality and joint weak stationarity giveConsequently the spectral matrix $F(\lambda )=\left(\begin{array}{cc}{f}_{XX}& f_{XY}\\ f_{YX}& f_{YY}\end{array}\right)$ is a Hermitian matrix. It is positive semidefinite, because its quadratic forms are nonnegative spectral densities of complex linear combinations of the jointly stationary series. Its nonnegative determinant therefore yields $|f_{XY}(\lambda ){|}^2\le f_{XX}(\lambda )f_{YY}(\lambda )$.

The imaginary part records directional lag information. For white noise $X_t$ of variance $v$ and $Y_t=X_{t-1}$, only ${\gamma }_{XY}(1)=v$ is nonzero. Hence $f_{XY}(\lambda )=v{e}^{-i\lambda }/(2\pi )$, although both individual spectral densities equal $v/(2\pi )$. The cross-spectrum is generally complex, rather than real and even like a real series' own spectral density.