Analytic Dependence of the Lyapunov Moment Function and the Projective Stationary Measure for Random Matrix Products

We consider the product of i.i.d. random matrices sampled according to a probability measure $μ$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of the Lyapunov moment function and the unique stationary measure on the projective space with respect to $μ$ in the total variation topology. As a consequence, we obtain the analyticity of the asymptotic variance and all higher-order Lyapunov moments.