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The power spectral density (PSD) of any time-dependent stochastic processes$X_t$ is a meaningful feature of its spectral content. In its text-bookdefinition, 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 asan ensemble-averaged property taken in the limit $T \to \infty$. A legitimatequestion is what information on the PSD can be reliably obtained fromsingle-trajectory experiments, if one goes beyond the standard definition andanalyzes the PSD of a \textit{single} trajectory recorded for a \textit{finite}observation time $T$. In quest for this answer, for a $d$-dimensional Brownianmotion we calculate the probability density function of a single-trajectory PSDfor arbitrary frequency $f$, finite observation time $T$ and arbitrary number$k$ of projections of the trajectory on different axes. We show analyticallythat the scaling exponent for the frequency-dependence of the PSD specific toan ensemble of Brownian motion trajectories can be already obtained from asingle trajectory, while the numerical amplitude in the relation between theensemble-averaged and single-trajectory PSDs is a fluctuating property whichvaries from realization to realization. The distribution of this amplitude iscalculated exactly and is discussed in detail. Our results are confirmed bynumerical simulations and single particle tracking experiments, with remarkablygood agreement. In addition we consider a truncated Wiener representation ofBrownian motion, and the case of a discrete-time lattice random walk. Wehighlight some differences in the behavior of a single-trajectory PSD forBrownian motion and for the two latter situations. The framework developedherein will allow for meaningful physical analysis of experimental stochastictrajectories.

Power spectral density of a single Brownian trajectory: What one can and cannot learn from it