  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 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.
