Theoretical contributions to Monte Carlo methods, and applications to Statistics.

Summary The first part of this thesis concerns the inference of non-standardized statistical models. We study two inference methods based on random sampling: Monte-Carlo MLE (Geyer, 1994), and Noise Contrastive Estimation (Gutmann and Hyvarinen, 2010). The latter method was supported by a numerical justification of better stability, but no theoretical results had yet been proven. We prove that Noise Contrastive Estimation is more robust to the choice of the sampling distribution. We evaluate the gain in accuracy as a function of the computational budget. The second part of this thesis concerns approximate random sampling for high dimensional distributions. The performance of most sampling methods deteriorates rapidly as the dimension increases, but several methods have proven their efficiency (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). Following some recent work (Eberle et al., 2017 . Cheng et al., 2018), we study some discretizations of a process known as kinetic Langevin diffusion. We establish explicit convergence speeds to the sampling distribution, which have a polynomial dependence in the dimension. Our work improves and extends the results of Cheng et al. for log-concave densities.