  We establish large deviation type estimates for i.i.d. products of two
dimensional random matrices with finitely supported probability distribution.
The estimates are stable under perturbations and require no irreducibility
assumptions. In consequence, we obtain a uniform local modulus of continuity
for the corresponding Lyapunov exponent regarded as a function of the support
of the distribution. This in turn has consequences on the modulus of continuity
of the integrated density of states and on the localization properties of
random Jacobi operators.
