  The power spectral density (PSD) of any time-dependent stochastic processes
$X_t$ is a meaningful feature of its spectral content. In its text-book
definition, the PSD is the Fourier transform of the covariance function of
$X_t$ over an infinitely large observation time $T$, that is, it is defined as
an ensemble-averaged property taken in the limit $T \to \infty$. A legitimate
question is what information on the PSD can be reliably obtained from
single-trajectory experiments, if one goes beyond the standard definition and
analyzes the PSD of a \textit{single} trajectory recorded for a \textit{finite}
observation time $T$. In quest for this answer, for a $d$-dimensional Brownian
motion we calculate the probability density function of a single-trajectory PSD
for arbitrary frequency $f$, finite observation time $T$ and arbitrary number
$k$ of projections of the trajectory on different axes. We show analytically
that the scaling exponent for the frequency-dependence of the PSD specific to
an ensemble of Brownian motion trajectories can be already obtained from a
single trajectory, while the numerical amplitude in the relation between the
ensemble-averaged and single-trajectory PSDs is a fluctuating property which
varies from realization to realization. The distribution of this amplitude is
calculated exactly and is discussed in detail. Our results are confirmed by
numerical simulations and single particle tracking experiments, with remarkably
good agreement. In addition we consider a truncated Wiener representation of
Brownian motion, and the case of a discrete-time lattice random walk. We
highlight some differences in the behavior of a single-trajectory PSD for
Brownian motion and for the two latter situations. The framework developed
herein will allow for meaningful physical analysis of experimental stochastic
trajectories.
