Universal quantization of parametric sources has redundancy k/2 logn/n

Let {X_i}~P_θ, Θ∈Λ⊆IR^k. Rissanen has shown that there exist universal noiseless codes for {Xi} with per-letter rate redundancy as low as k/2 logn/n, where n is the blocklength and k is the number of source parameters. We derive an analogous result for universal quantization: for any given La-grange multiplier λ>0, there exist universal fixed-rate and variable-rate quantizers with per-letter Lagrangian redundancy (i.e., distortion redundancy plus λ times the rate redundancy) as low as λk/2 logn/n