论文标题
大量高斯时间序列测试独立性的大型随机矩阵方法
Large random matrix approach for testing independence of a large number of Gaussian time series
论文作者
论文摘要
复杂的高斯高维高度时间序列$（\ y_n）_ {n \ in \ mathbb {z}} $在a a asymptigon $ n下，$ n的$ n $ n $ n $ n $ n $ n $ n $ n $ n $ \ y $和估算值的平滑跨度以相同的速率增长到无穷大，以$ \ frac {m} {n} {n} \ rightarrow 0 $。已经确定，在每个频率下，估计的光谱相干矩阵与独立$ \ MATHCAL {n} _ {\ MATHBB {C}}（0，\ i_m）$分布序列的样本协方差矩阵接近，及其经验eigenvalue分布朝向MARCENKOCO，这可以得出结论，每个LSS都具有可以明确评估的确定性行为。使用浓度不等式，可以表明，每个LS偏离其确定性近似的频率上，上超级的数量级为$ \ frac {1} {1} {m} {m} {m} + \ frac {\ sqrt {\ sqrt {m}}} {是样本量。数值模拟支持我们的结果。
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.
