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Regular version of the site
Of all publications in the section: 9
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Article
Naumov A., Spokoiny V., Ulyanov V. V. Probability Theory and Related Fields. 2019. Vol. 174. No. 3-4. P. 1091-1132.

Let X_1, ... ,X_n be i.i.d. sample in R^p with zero mean and the covariance matrix S. The problem of recovering the projector onto the eigenspace of S from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector P_r on the subspace of the r-th eigenvalue and its empirical counterpart \hat{P}_r in terms of the effective trace of S. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector P_r from the given data. This procedure does not rely on the asymptotic distribution of || P_r - \hat{P}_r ||_2 and its moments, it applies for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples. 

Added: Oct 15, 2018
Article
Ahidar-Coutrix A., Le Gouic T., Paris Q. Probability Theory and Related Fields. 2020.

This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the 2-Wasserstein space.

Added: Nov 14, 2019
Article
Loubes J., Le Gouic T. Probability Theory and Related Fields. 2017. P. 901-917.

Based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random probabilities defined on a geodesic space (Ed). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the probability measures or of a growing set of probability measures.

Added: Oct 13, 2018
Article
Borodin A., Olshanski G. Probability Theory and Related Fields. 2009. Vol. 144. No. 1. P. 281-318.
Added: Feb 25, 2013
Article
Mariani M. Probability Theory and Related Fields. 2010. Vol. 147. No. 3–4. P. 607-648 .

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.

Added: Dec 4, 2017
Article
Konakov V., Mammen E. Probability Theory and Related Fields. 2000. Vol. 117. No. 4. P. 551-587.

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved.

Added: Oct 15, 2012
Article
Shaposhnikov S., Bogachev V., Roeckner M. Probability Theory and Related Fields. 2014. Vol. 158. P. 465-476.

We prove absolute continuity of space-time probabilities satisfying certain parabolic inequalities for generators of diffusions with jumps. As an application, we prove absolute continuity of transition probabilities of singular diffusions with jumps under minimal conditions that ensure absolute continuity of the corresponding diffusions without jumps.

Added: Oct 15, 2014
Article
Konakov V., Mammen E. Probability Theory and Related Fields. 2009. Vol. 143. No. 1. P. 137-176.

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points. 

Added: Dec 4, 2012
Article
Kolesnikov A. Probability Theory and Related Fields. 2008. No. 140 (1-2). P. 1-17.
Added: Mar 23, 2011