Universality and rates of convergence in lossy source coding

The authors show that without knowing anything about the statistics of a bounded real-valued memoryless source, it is possible to construct a sequence of codes, of rate not exceeding a fixed number R>0, such that the per-letter sample distortion converges to the distortion-rate function D(R) with probability one as the length of the message approaches infinity. It is proven that the distortion converges to D(R) as √log log n /log n almost surely, where n is the length of the data to be transmitted