When optimal entropy-constrained quantizers have only a finite number of codewords

An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion Ed(X, Q(X)) subject to a constraint on the output entropy H(Q(X)). In general, such an optimal entropy-constrained quantizer may have a countably infinite number of codewords. In this short paper, we show that if the tails of the distribution of X are sufficiently light (with respect to the distortion measure), then the optimal entropy-constrained quantizer has only a finite number of codewords. In particular, for the squared error distortion measure, if the tails of the distribution of X are lighter than the tails of a Gaussian distribution, then the optimal entropy-constrained quantizer has only a finite number of codewords.