We study the rank of the instantaneous or spot covariance matrix $\Sigma_X(t)$ of a multidimensional continuous semi-martingale $X(t)$. Given high-frequency observations $X(i/n)$, $i=0,\ldots,n$, we test the null hypothesis $rank(\Sigma_X(t))\le r$ for all $t$ against local alternatives where the average $(r+1)$st eigenvalue is larger than some signal detection rate $v_n$. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of $\Sigma_X(t)$. We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates $v_n$. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

翻译：我们研究一个多维连续半对数半半对数的美元X(t)美元的瞬时或即时共差矩阵的等级。鉴于高频观测 $X(i/n)$,美元=0,美元=0,美元=ldots,n美元,我们测试所有美元均值的无效假设 美元(sgma_X(t))\le r美元),以当地替代物为条件,因为平均(r+1)$steigen值大于某些信号检测率 $v_n美元。一个主要问题是,当地共变差统计的内在平均偏差产生偏差,扭曲了等级统计。我们显示,偏差取决于规律性以及美元=0,美元=0,美元=ldots(t) 美元的光谱差。我们建立明确的矩阵扰动和浓度结果,提供非偏差的统一关键值和最佳信号检测率 $v_n。这导致通过顺序测试得出等级估计方法。对于某类随机波动模型,我们通过标准正态的硬度模型来确定数据驱动临界值临界值,我们通过高频度模拟政府数据应用。