This paper studies the asymptotic spectral properties of the sample covariance matrix for high dimensional compositional data, including the limiting spectral distribution, the limit of extreme eigenvalues, and the central limit theorem for linear spectral statistics. All asymptotic results are derived under the high-dimensional regime where the data dimension increases to infinity proportionally with the sample size. The findings reveal that the limiting spectral distribution is the well-known Marchenko-Pastur law. The largest (or smallest non-zero) eigenvalue converges almost surely to the left (or right) endpoint of the limiting spectral distribution, respectively. Moreover, the linear spectral statistics demonstrate a Gaussian limit. Simulation experiments demonstrate the accuracy of theoretical results.

翻译：本文研究高维成分数据样本协方差矩阵的渐近谱性质，包括极限谱分布、极值特征值的极限以及线性谱统计量的中心极限定理。所有渐近结果均在数据维度随样本量成比例增加的高维框架下推导得出。研究发现极限谱分布为著名的Marchenko-Pastur定律。最大（或最小非零）特征值分别几乎必然收敛于极限谱分布的右（或左）端点。此外，线性谱统计量呈现高斯极限。仿真实验验证了理论结果的准确性。