The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.

翻译：光谱统计(LSS) 平滑时期的光谱显示器的表面行为 {LSS) 平滑时期的光度估计值, 一个复杂的高斯高地高地高地高地时序的光谱一致性矩阵 $( y_ n)\ n\ n\ n\ in\ mathbb ⁇ $ 独立元件的光谱显示器行为 。 在这种制度下, 样本大小为$\ 美元= * 美元, 而其尺寸为$$ 美元 和测算器平滑的分布以同样的速度增长到无限。 这样可以得出这样的结论, 每一个LSS 的确定性行为 $ N\ N\ n\\ n}\ rightrow 0$ 。 在每种频率下, 估计的光谱一致性矩阵与独立的 $( mac{N_ brb} 美元分布序列相近, 其经验性电子价值分布与 Marcenko- Pastur 分布相近。 这可以得出结论, 每一个LSS 和 mr=r= m rol dal 其浓度的频率都显示其浓度的顺序。