We present an extension of the ergodic, mixing and Bernoulli levels of the ergodic hierarchy in dynamical systems, the information geometric ergodic hierarchy, making use of statistical models on curved manifolds in the context of information geometry. We discuss the 2×2 Gaussian Orthogonal Ensembles (GOE) within a 2D correlated model. For values of the correlation coefficient vanishingly small, we find that GOE belong to the information geometric (IG) mixing level having a maximum negative value of scalar curvature. Moreover, we propose a measure of distinguishability for the family of distributions of the 2D correlated model that results to be an upper bound of the IG correlation.Instituto de Física La PlataConsejo Nacional de Investigaciones Científicas y Técnicas