  The effect of a stochastic displacement field on a statistically independent
point process is analyzed. Stochastic displacement fields can be divided into
two large classes: spatially correlated and uncorrelated. For both cases exact
transformation equations for the two-point correlation function and the power
spectrum of the point process are found, and a detailed study of them with
important paradigmatic examples is done. The results are general and in any
dimension. A particular attention is devoted to the kind of large scale
correlations that can be introduced by the displacement field, and to the
realizability of arbitrary ``superhomogeneous'' point processes.
