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

Abstract

We discuss the statistical properties of a single-trajectory power spectral density $S(\omega ,\mathcal{T})$ of an arbitrary real-valued centered Gaussian process $X(t)$, where $\omega$ is the angular frequency and $\mathcal{T}$ the observation time. We derive a double-sided inequality for its noise-to-signal ratio and obtain the full probability density function of $S(\omega ,\mathcal{T})$. 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 of these 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.

10 pages

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Cited by in corpus (2)

• Spectral fingerprints of non-equilibrium dynamics: The case of a Brownian gyrator
• Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes