Sharp non-asymptotic Concentration Inequalities for the Approximation of the Invariant Measure of a Diffusion

For an ergodic Brownian diffusion with invariant measure $\nu$, we consider a sequence of empirical distributions ($\nu$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($\gamma$n) n$\ge$1 along an adapted regular enough class of test functions f such that f --$\nu$(f) is a coboundary of the infinitesimal generator A. Denote by $\sigma$ the diffusion coefficient and $\Phi$ the solution of the Poisson equation A$\Phi$ = f -- $\nu$(f). When the square norm of |$\sigma$ * $\Phi$| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizations of $\nu$n(f) -- $\nu$(f). Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pag{\`e}s in [LP02], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. We provide as well a Slutsky like Theorem, for practical applications, where the deviation bounds are also asymptotically independent of the corresponding Poisson problem. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting

Concentration of empirical distribution functions with applications to non-i.i.d. models

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A concentration inequality for interval maps with an indifferent fixed point

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.Comment: 26 pages, submitte