Estimation of autocovariance matrices for high dimensional linear processes

Abstract

AbstractIn this paper under some mild restrictions upper bounds on the rate of convergence for estimators of $$p\times p$$ p × 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 $$p={\mathcal {O}}\left( n^{\gamma /2}\right) $$ p = O n γ / 2 for some $$\gamma >1$$ γ > 1 , and in the general case when $$ p^{2/\beta }(n^{-1} \log p)^{1/2}\rightarrow 0$$ p 2 / β ( n - 1 log p ) 1 / 2 → 0 for some $$\beta >2$$ β > 2 as $$ p=p(n)\rightarrow \infty $$ p = p ( n ) → ∞ and the sample size $$n\rightarrow \infty $$ n → ∞ . 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.