The Kullback-Leibler divergence rate of two finite or countable ergodic Markov chains is well-known to exist and have an explicit expression as a function of the transition matrices of the chains, allowing access to classical tools for applications, such as minimization under constraints or projections on convex sets. The existence of Rényi divergence rates of ergodic Markov chains has been established in [5], [12]; here we establish explicit expressions for them and prove some properties of the resulting measures of discrepancy between stochastic matrices, opening the way to applications.

We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, the estimates of φ-mutual informations, associated to φ-divergences between a joint distribution and the product distribution of its marginals, are derived through the dual representation of φ-divergences. The asymptotic properties of the proposed estimates are established, including consistency, asymptotic distributions, and large deviations principle. The obtained tests of independence are compared via their relative asymptotic Bahadur efficiency and numerical simulations. It follows that the proposed semiparametric mutual information test is the optimal one. On the other hand, the proposed approach provides a new method for estimating the mutual information in a semiparametric setting, as well as a model selection procedure in a large class of dependence models, including semiparametric copulas.