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 Q_n of increasing dimension n and fixed rate R such that the mean squared distortion Δ(Q_n) is bounded as Δ(Q_n )=D(R)+O(√(log n/n)), 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 (1968) on the transmission of a quantized Gaussian source over a memoryless channel. In addition we are able to obtain a rate of convergence result for universal lossy source coding