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Inference for High-Dimensional Sparse Spectral Precision Matrices

Gaussian graphical models in the spectral domain offer a principled approach for recovering conditional dependence structures in stationary high-dimensional time series. Inference on the spectral precision matrix at a fixed frequency enables tests of frequency-specific conditional associations among time series components. The problem is challenging because finite-sample discrete Fourier transforms induce truncation and smoothing biases, while the complex-valued nature of the spectral precision matrix complicates high-dimensional variance estimation, rendering methods for i.i.d. samples not directly applicable. Existing approaches do not provide full likelihood-based inference for the discrete Fourier transforms. We propose a high-dimensional inference framework for sparse spectral precision matrices using the full likelihood of neighboring discrete Fourier transforms. We construct a debiased complex graphical lasso estimator at any fixed frequency. Using asymptotic theory for quadratic forms of multivariate time series, we establish its asymptotic normality and construct entry-wise consistent covariance estimators by aggregating information across neighboring frequencies. The key theoretical contribution is the simultaneous control of regularization, finite-sample truncation, and smoothing biases, enabling valid inference. Simulation studies show reliable coverage away from zero frequency and improved detection power over the benchmark, with false discovery rates near the desired level.

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