Study of velocity centroids based on the theory of fluctuations in
  position-position-velocity space
  We study possibility of obtaining velocity spectra by studying turbulence in
an optically thick medium using velocity centroids (VCs).We find that the
regime of universal, i.e. independent of underlying turbulence statistics,
fluctuations discovered originally within the velocity channel analysis (VCA)
carries over to the statistics of VCs. In other words, for large absorptions
the VC lose their ability to reflect the spectra of turbulence. Combining our
present study with the earlier studies of centroids in Esquivel & Lazarian, we
conclude that centroids are applicable for studies subsonic/transsonic
turbulence for the range of scales that is limited by the absorption effects.
We also consider VCs based on absorption lines and define the range of their
applicability. We address the problem of analytical description of spectra and
anisotropies of fluctuations that are available through studies using VC. We
obtain spectra and anisotropy of VC fluctuations arising from Alfv\'en, slow
and fast modes that constitute the compressible MHD cascade to address the
issue of anisotropy of VC statistics, and show how the VC anisotropy can be
used to find the media magnetization as well as to identify and separate
contributions from Alfv\'en, slow and fast modes. Our study demonstrates that
VCs are complementary to the tools provided by the VCA. In order to study
turbulent volume for which the resolution of single dish telescopes is
insufficient, we demonstrate how the studies of anisotropy can be performed
using interferometers.
