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