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Let G := (X1,…,Xn) be a vector of standard Gaussian random variables. For a function H : ℝ → ℝ, we consider the normalized weighted H-sum of G, and under suitable hypotheses on G and H, we prove a Berry–Esséentype bound for it. We also prove a functional central limit theorem for a partial-sum process corresponding to such sums. The proof of the former theorem is based on a bound of the Kolmogorov...
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