  We study random products of matrices in SL_2(C) from the point of view of
holomorphic dynamics. For non-elementary measures with finite first moment we
obtain the exponential convergence towards the stationary measure in Sobolev
norm. As a consequence we obtain the exponentially fast equidistribution of
forward images of points towards the stationary measure. We also give a new
proof of the Central Limit Theorem for the norm cocycle under a second moment
condition, originally due to Benoist-Quint, and obtain some general regularity
results for stationary measures.
