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

@article{Krapf2018PowerSD,
  title={Power spectral density of a single Brownian trajectory: what one can and cannot learn from it},
  author={Diego Krapf and Enzo Marinari and Ralf Metzler and Gleb Oshanin and Xinran Xu and Alessio Squarcini},
  journal={New Journal of Physics},
  year={2018},
  volume={20}
}
The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit T → ∞ . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard… 
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