In this paper under some mild restrictions upper bounds on the rate of convergence for estimators of \(p\times p\) autocovariance and precision matrices for high dimensional linear processes are given. We show that these estimators are consistent in the operator norm in the sub-Gaussian case when \(p={\mathcal {O}}\left( n^{\gamma /2}\right) \) for some \(\gamma >1\), and in the general case when \( p^{2/\beta }(n^{-1} \log p)^{1/2}\rightarrow 0\) for some \(\beta >2\) as \( p=p(n)\rightarrow \infty \) and the sample size \(n\rightarrow \infty \). In particular our results hold for multivariate AR processes. We compare our results with those previously obtained in the literature for independent and dependent data. We also present non-asymptotic bounds for the error probability of these estimators.

本文在一些适度的限制下，给出了\（p \ times p \）自协方差估计量的收敛速度的上限，并给出了高维线性过程的精确矩阵。我们表明，这些估计是在亚高斯情况下，操作规范一致时\（P = {\ mathcal {Ø}} \左（N ^ {\伽马/ 2} \右）\）对一些\（\ γ> 1 \），通常情况下，对于某些\（\ beta，当\（p ^ {2 / \ beta}（n ^ {-1} \ log p）^ {1/2} \ rightarrow 0 \）> 2 \）为\（p = p（n）\ rightarrow \ infty \）和样本大小\（n \ rightarrow \ infty \）。特别是我们的结果适用于多元AR流程。我们将我们的结果与先前文献中获得的独立和相关数据进行比较。我们还为这些估计量的错误概率提供了非渐近边界。