Asymptotic distribution of the errors in scalar and vector quantizers
High-rate (or asymptotic) quantization theory has found formulas for the average squared length (more generally, the qth moment of the length) of the error produced by various scalar and vector quantizers with many quantization points. In contrast, this paper finds an asymptotic formula for the probability density of the length of the error and, in certain special cases, for the probability density of the multidimensional error vector, itself. The latter can be used to analyze the distortion of two-stage vector quantization. The former permits one to learn about the point density and cell shapes of a quantizer from a histogram of quantization error lengths. Histograms of the error lengths in simulations agree well with the derived formulas. Also presented are a number of properties of the error density, including the relationship between the error density, the point density, and the cell shapes, the fact that its qth moment equals Bennett's integral (a formula for the average distortion of a scalar or vector quantizer), and the fact that for stationary sources, the marginals of the multidimensional error density of an optimal vector quantizer with large dimension are approximately i.i.d. Gaussian.