The distortion of vector quantizers trained on n vectors decreases to the optimum as Op(1/n)

Recently it has been experimentally observed that the expected squared error of a fixed-rate vector quantizer trained on n vectors decreases to the expected squared error of the optimal fixed-rate vector quantizer as roughly O(1/n). We confirm this observation theoretically, using Pollard´s (1982) result that the codewords of a fixed-rate vector quantizer trained on n independent vectors converge to the codewords of the unique optimal vector quantizer according to a central limit theorem