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.

翻译：频域中的高斯图模型为恢复平稳高维时间序列的条件依赖结构提供了基本原理。对固定频率下的谱精度矩阵进行推断，能够检验时间序列分量间特定频率的条件关联。该问题具有挑战性，因为有限样本离散傅里叶变换引入了截断和平滑偏差，而谱精度矩阵的复值特性使高维方差估计复杂化，导致独立同分布样本的方法无法直接适用。现有方法无法为离散傅里叶变换提供完整的基于似然的推断。我们提出了一种利用相邻离散傅里叶变换完全似然的高维推断框架。在任意固定频率下，我们构建了一个去偏的复图 lasso 估计量。利用多元时间序列二次型的渐近理论，我们证明了其渐近正态性，并通过聚合相邻频率的信息构建了逐项一致的协方差估计量。关键的理论贡献在于同时控制正则化、有限样本截断和平滑偏差，从而实现了有效的推断。模拟研究表明，在远离零频率处具有可靠的覆盖率，且检测能力优于基准方法，错误发现率接近预期水平。