Abstract: This paper is in order to extend the memoryless discrete source coding theorem to the case of nonhomogeneous Markov chain which leads to a greater application range. Firstly, we establish a strong law of large numbers of a delayed nonhomogeneous Markov chain by the classical Borel-Cantelli lemma. Then, we apply an independent random source to approximate the nonhomogeneous Markov source in order to obtain the general coding theorem of the nonhomogeneous Markov source. Finally, with the help of the general coding theorem of nonhomogeneous Markov source, we propose the a numerical method of the lowest fault tolerance rate in batch data hypothesis test.

Key words: nonhomogeneous markov source, coding theorem, hypothetical test, Borel-Cantelli lemma, unanimously bounded