Stationary distribution of product of matrices with random coefficients

The study devoted to the probabilistic product properties for a large number of independent equally distributed random matrices is based on a number of results obtained by H. Furstenberg (1963). Particularly, he proved the ergodicity of Markov chains caused by the action of random matrices on some compact uniform subspace of the group of matrices W called the boundary of this group. The stationary distribution of this chain (the invariant probability measure) defines the characteristics of the limiting behavior of the matrix product. Up to now, this measure was found only in simple cases. As an example, we consider the fundamental matrix of the Jacobi equation with random curvature to compute the invariant measure. Using this measure, we compute the Lyapunov exponent and the growth rate of statistical moments of the Jacobi field. Our results are compared with the results obtained previously by the application of the Monte Carlo method; a high degree of coincidence is observed.