This paper deals with a statistical approach for description of the laser field interaction with underdense plasmas and modification of the laser beam temporal coherence during its propagation through a plasma at power well below the filamentation threshold. The main properties of the plasma density perturbations driven by a randomized laser beam are derived from a stochastic wave equation. The laser spectral and angular broadening is shown to occur on a distance that depends essentially on the ratio of the average power in a speckle to the critical power for filamentation. The coherence time of the transmitted light is reduced to the plasma acoustic time of response to the laser. It is typically a few picoseconds. Dedicated diagnostics have been developed for the interaction code PARAX in order to analyze the laser and plasma statistical properties. The effect of the plasma length on the transmitted light coherence is found to be in good agreement with theoretical predictions. Forward stimulated Brillouin scattering is shown to play a key role in the laser coherence loss in this low-intensity regime. The limitations of the analytical model are discussed in terms of the deviation of the electric field distribution from the Gaussian statistics and creation of density-electric field correlations. This regime of laser induced incoherence is especially interesting in that the associated angular broadening is not as deleterious as observed for higher intensities. Moreover, beam smoothing can be achieved in low-density plasmas where energy losses due to absorption and backscattering are not too important.