Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes

We discuss the statistical properties of a single-trajectory power spectral density S(omega, tau) of an arbitrary one-dimensional real-valued centered Gaussian process X(t), where w is the angular frequency and tau the observation time. We derive a double-sided inequality for its noise-to-signal ratio and obtain the full probability density function of S(omega, tau). Our findings imply that the fluctuations of S(omega, tau) exceed its average value mu(omega, tau). This implies that using mu(omega, tau) to describe the behavior of these processes can be problematic. We finally evaluate the typical behavior of S(omega, tau) and find that it deviates markedly from the average mu(omega, tau) in most cases.