The spectrum of the weighted sample covariance shows a asymptotic non random behavior when the dimension grows with the number of samples. In this setting, we prove that the asymptotic spectral distribution $F$ of the weighted sample covariance has a continuous density on $\mathbb{R}^*$. We address then the practical problem of numerically finding this density. We propose a procedure to compute it, to determine the support of $F$ and define an efficient grid on it. We use this procedure to design the $\textit{WeSpeR}$ algorithm, which estimates the spectral density and retrieves the true spectral covariance spectrum. Empirical tests confirm the good properties of the $\textit{WeSpeR}$ algorithm.

翻译：当维度随样本数量增长时，加权样本协方差矩阵的谱呈现渐近非随机特性。在此设定下，我们证明了加权样本协方差矩阵的渐近谱分布 $F$ 在 $\mathbb{R}^*$ 上具有连续密度。随后，我们探讨了数值求解该密度的实际问题，提出了一种计算该密度、确定 $F$ 的支撑集并在其上定义高效网格的流程。基于此流程，我们设计了 $\textit{WeSpeR}$ 算法，用于估计谱密度并恢复真实谱协方差矩阵的谱。实证测试验证了 $\textit{WeSpeR}$ 算法的优良性能。