  Gravitational redshift as a relativistic effect in cosmological objects is
investigated. Possible signatures of the gravitational redshift in measurements
of satellite galaxies in clusters of galaxies, intracluster gas, as well as
galaxies associated with voids are investigated by developing simple
theoretical models. In the analysis of the gravitational redshift of satellite
galaxies, we develop a very simple analytic model for satellite galaxies
virialised in halos, which enables us to evaluate the signals depending on the
properties of the halo occupation distribution of galaxies. We obtain results
consistent with recent previous results, though our results are restricted to
the satellite galaxies inside the virial radius. In the analysis of
intracluster gas, we develop a simple analytic model including the effect of
random motions of gases, which are assumed to generate nonthermal pressure. We
demonstrate a possible contribution of the random motions of gases to the
gravitational redshift. We also investigate a possible signature of the
gravitational redshift in measurements of galaxies associated with voids, for
the first time as far as we know, by utilising a simple analytic model. We show
that the second-order Hubble term, which appears in the expansion of the scale
factor around the centre of a void, may make a significant contribution
depending on the way the galaxy samples are analysed.
