Оценивание диффузионных матриц для высокоразмерных процессов Леви и деконволюция для распределений с разреженными ковариационными матрицами

In the first part of the talk we consider the problem of estimating the diffusion matrix Σ of a high-dimensional, possibly time-changed Levy process based on discrete observations of the process with a fixed distance, where 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. In the second part we study the estimation of the covariance matrix Σ of a p-dimensional normal random vector based on n independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of Σ. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates.
The talk is based on the joint work with M. Trabs (Uni Hamburg) and A. Tsybakov (CREST ENSAE).