Asymptotic non-linear shrinkage formulas for weighted sample covariance

We compute asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators for weighted sample covariances, in the spirit of Ledoit and P\'ech\'e. We detail explicitly the formulas for exponentially-weighted sample covariances. Those new tools pave a way for applying non-linear shrinkage methods on weighted sample covariance. We show experimentally the performance of the asymptotic shrinkage formulas. Finally, we test the robustness of the theory to a heavy-tailed distributions.