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Journal of Probability and Statistics
Volume 2013 (2013), Article ID 645151, 6 pages
The Central Limit Theorem for th-Order Nonhomogeneous Markov Information Source
College of Mathematics and Information Science, Wenzhou University, Zhejiang 325035, China
Received 22 August 2013; Accepted 17 October 2013
Academic Editor: Shein-chung Chow
Copyright © 2013 Huilin Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  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  and Sarymsakov . Based on Dobrushin's work, Sethuraman and Varadhan  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 matrix whose elements are , and we will introduce the ergodic -coefficient equal to
Now we define another stochastic matrix as follows: where is called the m-dimensional stochastic matrix determined by the th-order transition matrix.
Let be the number of in the sequence of ; that is,
Lemma 1 (see ). Let be an th-order nonhomogeneous Markov information source which is taking values in state space with initial distribution of (3) and th-order transition probability matrices (5). is defined as (13). Let be another m-order transition matrix, and let be the m-dimensional stochastic matrix determined by the th-order transition matrix , that is, . Suppose that Then one has
2. Statement of the Main Result
Let be any Borel function defined on product space . Denote where Obviously, is a martingale, so that is the associated martingale difference sequence. Denote
Our main result is describing conditions on and under in which the central limit theorem holds for the stochastic sequence .
Theorem 2. Let be an th-order nonhomogeneous Markov information source which is taking values in state space with initial distribution of (3) and th-order transition probability matrices (5). Let be an -th order transition matrix. Let be any function defined on the state space and let be defined as (13). If (14) holds and the sequence of -coefficients for the th-order stochastic matrices satisfies that then one has where denotes the convergence in distribution and
Remark 3. The sequence is said to converge in the Cesàro sense to constant matrix if (14) holds.
3. Proof of Theorem 2
Let be a probability space and let be a sequence of random variables which is defined on . Let be an increasing sequence of -fields of sets. Now let be a sequence of martingale, so that is a martingale difference. is a trivial field. For , denote
Lindeberg Condition. For , where denotes the index function.
In our proof, we will use the central limit theorem of martingale sequences as the technical tool.
Lemma 4 (see ). Suppose that the sequence of martingale satisfies the following condition: Moreover, if the Lindeberg condition holds, then one has where and denote convergence in probability and in distribution, respectively.
Theorem 5. Let be an -order nonhomogeneous Markov information source which is taking values in state space with initial distribution of (3) and th-order transition probability matrices (5). Let be any function defined on the state space . Suppose that the function satisfies condition (21). Let be defined as (16). If (14) holds, then where denotes the convergence in distribution.
Proof of Theorem 5. Noting that by using the property of the conditional expectation and Markov property, it follows from (17) that
Noting that, on the one hand,
which tends to zero as tends to infinity by using (14). Thus we have
where the third equation holds because of (15). Combining (29) and (32), we get
On the other hand, let us come to compute the limit of as tends to infinity. By using (14) again, we have
Thus by Lemma 1, we easily arrive at
Combining (28), (33), and (35), we arrive at which implies that Note that Since is a finite set, then the random sequence is uniformly integrable. Combining above two facts, we arrive at
It follows that where . On the other hand, similar to the analysis of inequality (38), we also have that is uniformly integrable, so that which implies that the Lindeberg condition holds, and then we can easily get our conclusion by using Lemma 4.
Now let us come to prove our main result of Theorem 2.
Proof of Theorem 2. Note that
Denote and . Let us come to estimate the upper bound of . In fact, it follows from the C-K formula where . By using condition (19), we get Then, by using (27), (42), and (45), we can arrive at our conclusion (20). Thus the proof of Theorem 2 is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China no. 11201344.
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