| Probability theory is a subject which concerns the study statistical laws for different kinds of random phenomenon, paticularly, the limit behavior when the abserving times tend to infinity. Hence the strong limit theorems consist of an important part of probability theory. In 1960's, the limit theorem for the sequences of independent random variables has been well established. Since then, the limit theorems for mixing sequences of random variables and correlated sequences of random variables have been developed greatly. Many chinese researchers have contributed outstandingly in this field. Their influential works have been international recognized (Cf[43, 76, 108, 81, 77, 82] ). There are vast number of references upon limit theorems in literature, of which the classical ones (Cf[14, 13, 70, 28, 79]). Some recently updated references are [31, 4, 32, 69, 16, 11]. The entropy theorem in information theory, which is of core interst in this thesis, is also frequently as the Shannon-McMillan theorem or asymptotic equipartition property (AEP). It is fundamental theorem in information theory which lays the foundation of almost all the coding theorems. The most recent development of entropy theorem could be found in [26].Let {Xn, n ≥ 0} be a stochastic sequence. IfE[f(Xn+l)|X0,... ,Xn] = E[f(Xn+l)|Xn] a.s., (-1.0.3)where f is a bounded function, {Xn, n ≥ 0} will be called a Markov chain. If E[f (Xn+1)|Xn] are dependent on n, {Xn,n≥ 0} will be called a nonhomogeneous Markov chain. If E[f(Xn+1)|Xn] are independent on n, {Xn,n ≥ 0} will be called a homogeneous Markov chain. If {Xn, n ≥ 0} take values in a finite set or a countable set, it will be called a finite Markov chain or countable Markov chain. If {Xn, n ≥ 0} take values in a general state space, it will be called a Markov chain...
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