Understanding the Dyer-Roeder approximation as a consequence of local
  cancellations of projected shear and expansion rate fluctuations
  It is shown with several concrete examples that the Dyer-Roeder approximation
is valid in spacetimes which fulfill the condition that fluctuations in the
expansion rate along a light ray locally cancels with the shear contribution to
the redshift. This is the case for standard cosmological scenarios including
perturbed FLRW spacetimes, N-body simulations and Swiss-cheese models. With
another concrete example it is then illustrated that it is possible to
construct statistically homogeneous and effectively statistically isotropic
cosmological models which do not fulfill the condition. In this case, the
Dyer-Roeder approximation is invalid. Instead, the mean redshift-distance
relation can be described using a relation based on the spatial averages of the
transparent part of the spacetime.
