The Erd(?)s-r(?)nyi Law And Stationary Finite States Markov Chain

This paper studies the ERDOS-RENYI Law for a single finite states irre-ducible aperiodic and stationary Markov Chain. combined with the theorem of the ERDOS-RENYI Law for two Markov Chains, these two theorems forms a basis from i.i.d chains to no i.i.d chains. The ERDOS-RENYI Law for a single stationary Markov Chain seems similar to the ERDOS-RENYI Law for two chains. The detail of the proof analyzes the upper bound and the lower bound, following the method of Pattern, Position analysis and the Borel-Cantelli lemma.For the convenience of calculating, we try to give a simple expression of H(a, [Π]). For finite states irreducible aperiodic Markov Chain, we give such a simple expression of H (a, [Π]), by using the large deviation principle and S anov's theorem. The simple expression of H (a, [Π]) will be of great help for the application of the the-orem. Give a short explain of the covariance of a Markov Chain. The conclusion is that the correlation is not monotone all the time.There are 3 experiments. The first and the second experiments generate Markov Chains by the help of MATLAB, and the result seems to be good. In the third experi-ment we get the data from NCBI, a series of mRNA,named Mus musculus Max protein (Max), transcript variant 1,2005bp. The result is good.The new features of this paper are as follows:●Consider the simplest and no i.i.d chain.●For a two-state and irreducible aperiodic Markov Chain, we give a simple ex-pression of the entropy H.●Analyze a series of mRNA, and get a good result.