L2L^2 Asymptotics for High-Dimensional Data

We develop an asymptotic theory for $L^2$ norms of sample mean vectors of high-dimensional data. An invariance principle for the $L^2$ norms is derived under conditions that involve a delicate interplay between the dimension $p$, the sample size $n$ and the moment condition. Under proper normalization, central and non-central limit theorems are obtained. To facilitate the related statistical inference, we propose a plug-in calibration method and a re-sampling procedure to approximate the distributions of the $L^2$ norms. Our results are applied to multiple tests and inference of covariance matrix structures.Comment: 3

Regularized estimation of linear functionals of precision matrices for high-dimensional time series

This paper studies a Dantzig-selector type regularized estimator for linear functionals of high-dimensional linear processes. Explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from i.i.d. samples to long-range dependent time series and from sub-Gaussian innovations to those with mild polynomial moments. It is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes. The Dantzig-selector estimator is applied to the sparse Markowitz portfolio allocation and the optimal linear prediction for time series, in which the ratio consistency when compared with an oracle estimator is established. The effect of dependence and innovation moment conditions is further illustrated in the simulation study. Finally, the regularized estimator is applied to classify the cognitive states on a real fMRI dataset and to portfolio optimization on a financial dataset.Comment: 44 pages, 4 figure