  We investigate the stochastic behavior of the single-trajectory spectral
density $S(\omega,\mathcal{T})$ of several Gaussian stochastic processes, i.e.,
Brownian motion, the Ornstein-Uhlenbeck process, the Brownian gyrator model and
fractional Brownian motion, as a function of the frequency $\omega$ and the
observation time $\mathcal{T}$. We evaluate in particular the variance and the
frequency-frequency correlation of $S(\omega,\mathcal{T})$ for different values
of $\omega$. We show that these properties exhibit different behaviors for
different physical cases and can therefore be used as a sensitive probe
discriminating between different kinds of random motion. These results may
prove quite useful in the analysis of experimental data.
