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We investigate the propagation of light rays in a clumpy universe constructed by a cosmological version of the post-Newtonian approximation. We show that the linear approximation to the propagation equations is valid in the region z1 even if the density contrast is much larger than unity. Based on a general order-of-magnitude statistical consideration, we argue that the linear approximation is still valid for z1. Then we give a general formula for the distance-redshift relation in a clumpy universe and derive an explicit expression for a simplified situation in which the effect of the gravitational potential of inhomogeneities dominates. In the light of the derived relation we discuss the validity of the Dyer-Roeder distance. Furthermore, we consider a simple model of an inhomogeneous universe and investigate statistical properties of light rays. We find that the result of this specific example also supports the validity of the linear approximation.
Futamase et al. (Sun,) studied this question.