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| SOME LIMIT PROPERTIES AND THE GENERALIZED AEP THEOREM FOR NONHOMOGENEOUS MARKOV CHAINS |
| Ping Hu,Zhongzhi Wang |
| (School of Math. $\&$ Physics Science and Engineering, AnHui University of Technology, Ma'anshan 243002, Anhui, PR China) |
| DOI： |
| Abstract: |
| Let $(\xi_n)_{n=0}^\infty$ be a Markov chain with the state space $\mathcal{X}=\{1,2,\cdots,b\}$, $(g_n(x,y))_{n=1}^\infty$ be functions defined on athcal{X}\times\mathcal{X}$, and $$F_{m_n,b_n}(\omega)=\frac{1}{b_n}\sums_{k=m_n+1}^{m_n+b_n} g_{k}(\xi_{k-1},\xi_{k}).$$ In this paper the limit properties of $F_{m_n,b_n}(\omega)$ and the generalized relative entropy density $f_{m_n,b_n}(\omega)=-(1/b_n)\log p(\xi_{m_n,m_n+b_n})$ are discussed, and some theorems on
a.s. convergence for $(\xi_n)_{n=0}^\infty$ and the generalized Shannon-McMillan (AEP) theorem on finite nonhomogeneous Markov chains are obtained. |
| Key words: AEP; nonhomogeneous Markov chains; limit theorem; generalized relative entropy density |