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 |(/spl part///spl part/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/sub n/}/sub n=1//sup /spl infin// of block codes, where C/sub n/ is an nth-order block code at the fixed rate R, and any /spl epsiv/>0, the redundancy D/sub n/(C/sub n/,p) of C/sub n/ for p is greater than or equal to |(/spl part///spl part/R)d(p,R)|(K-/spl epsiv/)ln n/2n for all p satisfying some regular conditions except a set whose volume goes to 0 as n/spl rarr//spl infin/. On the other hand, there exists a sequence {C/sub n/}/sub n=1//sup /spl infin// of block codes at the rate R such that for any p satisfying some regular conditions, the super limit of D/sub n/(C/sub n/,p)|(ln n/n) is less than or equal to |(/spl part///spl part/R)d(p,R)|K/2.