In this paper, we relate transport-entropy inequalities to the minimization of certain functionals defined on the space of probability measures. This approach leads in particular to a new proof of a result by Otto and Villani ( J. Funct. Anal. 173 (2000) 361–400) showing that the logarithmic Sobolev inequality implies Talagrand's transport inequality.
The estimation of the diffusion matrix Σ of a high-dimensional, possibly time-changed Levy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on Σ. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of Σ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.
We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant distribution νν of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions ff such that f−ν(f)f−ν(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when some suitable squared-norms of the diffusion coefficient also belong to this class. We apply these estimates to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.
In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.