This work develops non-asymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional Gaussian time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sparse. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model. Novel algorithms for the considered procedures are proposed, and a simulation study and a data application are also provided.

翻译：这项工作为估算长期差异矩阵及其反向,即所谓的精确矩阵,根据对依赖性结构的一般假设,包括长距离依赖性,为高斯高斯高斯高地时间序列,开发了非零星理论,用于估算长期差异矩阵及其反向,即所谓的精确矩阵。估计涉及正在对传统多变式本地Whittle天文测量仪的版本进行临界和惩罚的缩略技术。结果确保在双重零位系统中进行一致的估计,只要真实的模型参数稀少,就可以随着样本的大小而增加组成部分时间序列的数量。关键的技术结果是,本地的Whittle估计器在围绕真实模型参数的长期差异矩阵上的集中性不平等。特别是,它同时处理输入基本模型的记忆参数的估算。还提出了审议程序的新算法,并提供了模拟研究和数据应用。