  In this paper we consider the Ito-Schrodinger model for wave propagation in
random media in the paraxial regime. We solve the equation for the fourth-order
moment of the field in the regime where the correlation length of the medium is
smaller than the initial beam width. As applications we prove that the centered
fourth-order moments of the field satisfy the Gaussian summation rule, we
derive the covariance function of the intensity of the transmitted beam, and
the variance of the smoothed Wigner transform of the transmitted field. The
second application is used to explicitly quantify the scintillation of the
transmitted beam and the third application to quantify the statistical
stability of the Wigner transform.
