On the cost of finite block length in quantizing unbounded memoryless sources

The problem of fixed-rate block quantization of an unbounded real memoryless source is studied. It is proved that if the source has a finite sixth moment, then there exists a sequence of quantizers Qn of increasing dimension n and fixed rate R such that the mean squared distortion A(Q,) is bounded as A(Q,) < D(R) + O(dm), where D(R) is the distortion-rate function of the source. Applications of this result include the evaluation of the distortion redundancy of fixed-rate universal quantizers, and the generalization to the non-Gaussian case of a result of Wyner on the transmission of a quantized Gaussian source over a memoryless channel.