TY - JOUR
T1 - Some Limit Properties and the Generalized AEP Theorem for Nonhomogeneous Markov Chains
AU - Hu , Ping
AU - Wang , Zhongzhi
JO - Annals of Applied Mathematics
VL - 3
SP - 269
EP - 284
PY - 2022
DA - 2022/06
SN - 34
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/aam/20577.html
KW - AEP, nonhomogeneous Markov chains, limit theorem, generalized relative entropy density.
AB - <p style="text-align: justify;">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.</p>