The redundancy of source coding with a fidelity criterion .II. Coding at a fixed rate level with unknown statistics

The redundancy problem of universal lossy source coding at a fixed rate level is considered. Under some condition on the single-letter distortion measure, which implies that the cardinality K of the reproduction alphabet is not greater than the cardinality J of the source alphabet, it is shown that the redundancy of universally coding memoryless sources p by nth-order block codes of rate R goes like |(∂/∂R)d(p,R)|Kln n/2n+o(ln n/n) for all memoryless sources p except a set whose volume goes to 0 as the block length n goes to infinity, where d(p,R) denotes the distortion rate function of p. Specifically, for any sequence {C_n}_n=1^∞ of block codes, where C_n is an nth-order block code at the fixed rate R, and any ε>0, the redundancy D_n(C _n,p) of C_n for p is greater than or equal to |(∂/∂R)d(p,R)|(K-ε)ln n/2n for all p satisfying some regular conditions except a set whose volume goes to 0 as n→∞. On the other hand, there exists a sequence {C_n}_n=1^∞ of block codes at the rate R such that for any p satisfying some regular conditions, the super limit of D_n(C_n,p)|(ln n/n) is less than or equal to |(∂/∂R)d(p,R)|K/2