Some Limit Properties For The Functions Of Two Variables Of Nonhomogeneous Markov Chain Fields On Caylay Trees

Random fields on tree are applications on trees of theory of random process-a new math model ,which developed from coding and encoding probelm in information theory. Assuming there is one sequence {Xn}, whether the appearing frequency of state and state couple obeys the large number law is the key of a good coding and encoding method, so this domain is always being a researching emphases for most scholars. Recently, Professor Liu Wen and his associates study the strong law of large numbers for Markov chain fields on trees. But for the nonhomogeneous Markov chain fields ,they only study the even-odd Markov chain and non-symmetric Markov chain. However, this article goes in the normal nonhono-geneous Markov chain fields and gets a class of limit theorems for the functions of two variables of finite nonhomogeneous Markov chain fields. Some limit theorems on the frequences of states and ordered state couples are obtained by several corollaries. At last,we extend the Shannon-McMillan theorem to the nonhomogeneous Markov chain fields.