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This work develops non-asymptotic theory for estimation of the long-runvariance matrix and its inverse, the so-called precision matrix, forhigh-dimensional time series under general assumptions on the dependencestructure including long-range dependence. The estimation involves shrinkagetechniques which are thresholding and penalizing versions of the classicalmultivariate local Whittle estimator. The results ensure consistent estimationin a double asymptotic regime where the number of component time series isallowed to grow with the sample size as long as the true model parameters aresparse. The key technical result is a concentration inequality of the localWhittle estimator for the long-run variance matrix around the true modelparameters. In particular, it handles simultaneously the estimation of thememory parameters which enter the underlying model. Novel algorithms for theconsidered procedures are proposed, and a simulation study and a dataapplication are also provided.

Local Whittle estimation of high-dimensional long-run variance and precision matrices