Products of random matrices and random affine transformations

We consider a random walk by linear or affine transformations on Euclidean space, such that the subgroup generated by the support of the corresponding probability measure is Zariski dense. We show that the stationary measure is homogeneous at infinity. One tool is a spectral gap property for projective actions, based on simplicity of the dominant Lyapunov exponent for a product of Markov-dependant random matrices.