Inference for High-Dimensional Sparse Spectral Precision Matrices

Statistical inference for the spectral precision matrix at a given frequency allows us to assess frequency-specific conditional relationships among the components of a stationary multivariate time series. Compared with classical analogues, inference in the spectral domain is more challenging due to the absence of closed-form asymptotic variance expressions for complex-valued estimators, the limited asymptotic theory in high-dimensional settings, and the presence of truncation and smoothing biases in finite samples. We construct a debiased complex graphical lasso estimator at any fixed frequency by leveraging the full likelihood structure of neighboring discrete Fourier transforms. Using asymptotic distributions for bilinear forms of stationary multivariate time series, we establish the joint asymptotic normality of the real and imaginary parts of the debiased estimator. Our main theoretical contributions include deriving a closed-form asymptotic covariance matrix for the real and imaginary parts, establishing a central limit theorem for bilinear forms of the underlying time series, and controlling smoothing and truncation biases in covariance estimation to ensure valid inference. Simulation studies demonstrate reliable coverage and improved statistical power relative to the benchmark, while maintaining false discovery rates near the nominal level. An application to real fMRI data further reveals distinct patterns of functional connectivity across selected frequencies.