Fundamental limit and pointwise asymptotics of the Bayes code for Markov sources

This paper considers universal lossless variable-length source coding problem and deals with one of the fundamental limits and pointwise asymptotics of the Bayes code for stationary ergodic finite order Markov sources. As investigation of the fundamental limits, we show upper and lower bounds of the minimum rate such that the probability which exceeds it is less than ε ∈ (0, 1). Furthermore, we prove that the codeword length of the Bayes code satisfies the asymptotic normality (pointwise √n asymptotics) and the law of the iterated logarithm (pointwise √n log log n asymptotics), where n represents length of a source sequence and “log” is the natural logarithm.