Abstract
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.
Highlights
There have been developments in frequency domain methodologies, though these proposed methodologies lack theoretical justifications
In order to allow for high dimensions, a common practice is to exploit concentration inequalities, to provide fixed-sample results to control the differences between the sample and the population versions, and to use union bound arguments to derive desirable results
In order to bound the errors between the smoothed periodogram matrix f and the spectral density matrix f, we introduce a series of instrumental quantities, including an m-dependent series using conditional expectations, its truncated version which is truncated in magnitude by (log(n))2, and a centred version by subtracting the unconditional expectations
Summary
In order to study the theoretical performances, we adopt the functional dependency framework [45]. Let e0, {et, t ∈ Z} be i.i.d. random vectors. Et), i.e. replace e0 with e0 in Ft. Define Xt,i = gi(Ft) and θt,i = E|Xt,i − Xt,i|2 1/2,. Which is used as a dependency measure. It has been pointed out in [45] that a large family of common time series models, including linear processes, Volterra series, nonlinear transforms of linear processes and some nonlinear time series models, can be characterised by imposing proper conditions on (2.2). We use the sparsity definition in [9] to characterise the sparsity of precision matrices, i.e. let the parameter space Gq(cn,p, Mn,p) be denoted by. Where 0 ≤ q < 1, cn,p and Mn,p are potentially diverging as n and p grow
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