The redundancy of source coding with a fidelity criterion. 1. Known statistics

The problem of redundancy of source coding with respect to a fidelity criterion is considered. For any fixed rate R>0 and any memoryless source with finite source and reproduction alphabets and a common distribution p, the nth-order distortion redundancy D_n(R) of fixed-rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate R and the distortion rate function d(p,R) of the source p. It is demonstrated that for sufficiently large n, D_n(R) is equal to -(∂/∂R)d(p,R) ln n/2n+o(ln n/n), where (∂/∂R)d(p,R) is the partial derivative of d(p,R) evaluated at R and assumed to exist. For any fixed distortion level d>0 and any memoryless source p, the nth-order rate redundancy R_n(d) of coding at fixed distortion level d (or by using d-semifaithful codes) is defined as the minimum of the difference between the expected rate per symbol of any d-semifaithful code of length n and the rate-distortion function R(p,d) of p evaluated at d. It is proved that for sufficiently large n, R_n(d) is upper-bounded by ln n/n+o(ln n/n) and lower-bounded by In n/2n+o(In n/n). As a by-product, the lower bound of R_n(d) derived in this paper gives a positive answer to a conjecture proposed by Yu and Speed (1993)