  A useful approach for analysing multiple time series is via characterising
their spectral density matrix as the frequency domain analog of the covariance
matrix. When the dimension of the time series is large compared to their
length, regularisation based methods can overcome the curse of dimensionality,
but the existing ones lack theoretical justification. This paper develops the
first non-asymptotic result for characterising the difference between the
sample and population versions of the spectral density matrix, allowing one to
justify a range of high-dimensional models for analysing time series. As a
concrete example, we apply this result to establish the convergence of the
smoothed periodogram estimators and sparse estimators of the inverse of
spectral density matrices, namely precision matrices. These results, novel in
the frequency domain time series analysis, are corroborated by simulations and
an analysis of the Google Flu Trends data.
