Let $(ξ_n)^∞_{n=0}$ be a Markov chain with the state space $\chi = \{1, 2, · · · , b\},$ $(g_n(x, y))^∞_{n=1}$ be functions defined on $\chi \times \chi,$ and $$F_{m_n,b_n} (\omega) =\frac{1}{b_n}\sum\limits_{k=m_n+1}^{m_n+b_n}g_k(ξ_{k−1}, ξ_k).$$ In this paper the limit properties of $F_{m_n,b_n}(\omega)$ and the generalized relative entropy density $f_{m_n,b_n}(ω)=−(1/b_n){\rm log}p(ξ_{m_n,m_n+b_n})$ are discussed, and some
theorems on a.s. convergence for $(ξ_n)^∞_{n=0}$ and the generalized Shannon-McMillan
(AEP) theorem on finite nonhomogeneous Markov chains are obtained.
