Asymptotic spectrum of weighted sample covariance: another proof of spectrum convergence

We propose another proof of the high dimensional spectrum convergence of the weighted sample covariance, more concise and self-sufficient but with stronger, but reasonable assumptions. We explain and illustrates this theorem for different weight distributions and show how the spectrum behaves in finite samples with heavy tails. The general purpose is to provide a detailed introduction to the high dimensional spectrum of weighted sample covariance.