Weighted Means and Volatilities: Approximation, Convergence, and Sharp Bounds

We investigate the approximation of weighted means and volatilities from finitely many observations, with particular emphasis on exponentially decaying weights. We examine truncation, normalization, and recursive updating procedures, and extend the analysis to general systems of nonnegative weights. An exact mixture decomposition yields sharp uniform error bounds for the weighted mean, variance, and standard deviation in terms of the discarded weight mass. We further establish that the constants in these bounds are attained for every fixed geometric weighting scheme. The analysis extends to covariance matrices, providing sharp spectral-norm bounds and dimension-independent Frobenius-norm bounds. We also establish convergence under a finite weighted second-moment assumption and derive recursive formulas for time-varying forgetting factors. We distinguish universal sharpness from attainability for prescribed weighting schemes, and identify open questions concerning structurally constrained weights, sharp Frobenius-norm constants, and rounding-error analysis.