On the asymptotic distribution of the errors in vector quantization

In a recent paper, Lee and Neuhoff (see ibid., vol.42, no.2, p446-60, 1996) found an asymptotic formula for the distribution of the length of the errors produced by a vector quantizer with many quantization points. This distribution depends on the source probability density, the quantizer point density, and the quantizer shape profile. (The latter characterizes the shapes of the quantization cells as a function of position.) The purpose of this paper is to give a rigorous derivation of this formula by identifying precise conditions under which it is shown that if a sequence of vector quantizers with a given dimension and an increasing number of points has "specific" point densities and "specific" shape profiles converging to a "model" point density and a "model" shape profile, respectively, then the distribution of the length of the quantization error, suitably normalized, converges to the aforementioned formula, with the model point density and the model shape profile substituted.