The Central Limit Theorem for th-Order Nonhomogeneous Markov Information Source

We prove a central limit theorem for th-order nonhomogeneous Markov information source by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesàro sense. 1. Introduction Let be an arbitrary information source taking values on alphabet set with the joint distribution for . If is an th-order nonhomogeneous Markov information source, then, for , Denote where and are called the m-dimensional initial distribution and the th-order transition probabilities, respectively. Moreover, are called the th-order transition probability matrices. In this case, There are many of practical information sources, such as language and image information, which are often th-order Markov information sources and always nonhomogeneous. So it is very important to study the limit properties for the th-order nonhomogeneous Markov information sources in information theory. Yang and Liu [1] proved the strong law of large numbers and the asymptotic equipartition property with convergence in the sense of a.s. the th-order nonhomogeneous Markov information sources. But the problem about the central limit theorem for the th-order nonhomogeneous Markov information sources is still open. The central limit theorem (CLT) for additive functionals of stationary, ergodic Markov information source has been studied intensively during the last decades [2–9]. Nearly fifty years ago, Dobrushin [10, 11] proved an important central limit theorem for nonhomogeneous Markov information resource in discrete time. After Dobrushin's work, some refinements and extensions of his central limit theorem, some of which are under more stringent assumptions, were proved by Statuljavicius [12] and Sarymsakov [13]. Based on Dobrushin's work, Sethuraman and Varadhan [14] gave shorter and different proof elucidating more the assumptions by using martingale approximation. Those works only consider the case about th-order nonhomogeneous Markov chain. In this paper, we come to study the central limit theorem for th-order nonhomogeneous Markov information sources in Cesàro sense. Let be an th-order nonhomogeneous Markov information source which is taking values in state space with initial distribution of (3) and mth order transition probability matrices (5). Denote We also denote the realizations of by . We denote the th-order transition matrix at step by where . For an arbitrary stochastic square matrix whose elements are , we will set the ergodic -coefficient equal to where . Now we extend this idea to the th-order stochastic