ABSTRACT
This chapter introduces the key problem of structure formation theories: density fluctuations must be initially small enough not to perturb the overall scheme of the Friedmann-Robertson-Walker cosmology. It develops statistical tools to properly: set initial conditions to the gravitational instability equation; compare the theoretical predictions with the observations of the large-scale galaxy distribution or of the CMB angular anisotropies. The chapter introduces the basics of random field theories and shows their applications to the gravitational instability scenario and to large-scale redshift surveys. Bulk motions are important to provide a measure of the large-scale density fluctuations and to constrain theoretical models. However more information about peculiar motions can be derived by other sets of observables. In particular, the pairwise galaxy velocity dispersion can be measured from redshift surveys by modeling the distortion in redshift space of the galaxy-galaxy correlation function, an effect first discussed by Davis T. M. and Peebles P. J. E.
