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We discuss the statistical properties of a single-trajectory power spectraldensity $S(\omega,\mathcal{T})$ of an arbitrary real-valued centered Gaussianprocess $X(t)$, where $\omega$ is the angular frequency and $\mathcal{T}$ theobservation time. We derive a double-sided inequality for its noise-to-signalratio and obtain the full probability density function of$S(\omega,\mathcal{T})$. Our findings imply that the fluctuations of$S(\omega,\mathcal{T})$ exceed its average value $\mu(\omega,\mathcal{T})$.This implies that using $\mu(\omega,\mathcal{T})$ to describe the behavior ofthese processes can be problematic. We finally evaluate the typical behavior of$S(\omega,\mathcal{T})$ and find that it deviates markedly from the average$\mu(\omega,\mathcal{T})$ in most cases.

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