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

TL;DR

This paper analyzes the statistical fluctuations of single-trajectory spectral densities in Gaussian processes, revealing that their variability often exceeds the mean, which challenges the reliability of average-based descriptions.

Contribution

It derives the full probability density function and bounds for the noise-to-signal ratio of spectral densities in Gaussian processes, highlighting the limitations of mean-based analysis.

Findings

Fluctuations of spectral densities often surpass their mean values.

The full probability distribution of spectral densities is obtained.

Average spectral density may not accurately represent typical behavior.

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.