Journal of Probability and Statistics

Volume 2013 (2013), Article ID 645151, 6 pages

http://dx.doi.org/10.1155/2013/645151

Research Article

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

Huilin Huang

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.

Abstract.

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 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 [1]). 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 [15]). 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.

Before we prove our main result Theorem 2, we at first come to prove Theorem 5.

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

where

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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China no. 11201344.

References

W. G. Yang and W. Liu, “The asymptotic equipartition property for th-order nonhomogeneous Markov information sources,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3326–3330, 2004.

M. I. Gordin and B. A. Lifšic, “Central limit theorem for stationary Markov processes,” Doklady Akademii Nauk SSSR, vol. 239, no. 4, pp. 766–767, 1978.

Y. Derriennic and M. Lin, “The central limit theorem for Markov chains with normal transition operators, started at a point,” Probability Theory and Related Fields, vol. 119, no. 4, pp. 508–528, 2001.

Y. Derriennic and M. Lin, “The central limit theorem for Markov chains started at a point,” Probability Theory and Related Fields, vol. 125, no. 1, pp. 73–76, 2003.

M. I. Gordin and H. Holzmann, “The central limit theorem for stationary Markov chains under invariant splittings,” Stochastics and Dynamics, vol. 4, no. 1, pp. 15–30, 2004.

C. Kipnis and S. R. S. Varadhan, “Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions,” Communications in Mathematical Physics, vol. 104, no. 1, pp. 1–19, 1986.

M. Maxwell and M. Woodroofe, “Central limit theorems for additive functionals of Markov chains,” The Annals of Probability, vol. 28, no. 2, pp. 713–724, 2000.

M. Woodroofe, “A central limit theorem for functions of a Markov chain with applications to shifts,” Stochastic Processes and Their Applications, vol. 41, no. 1, pp. 33–44, 1992.

H. Holzmann, “Martingale approximations for continuous-time and discrete-time stationary Markov processes,” Stochastic Processes and Their Applications, vol. 115, no. 9, pp. 1518–1529, 2005.

R. Dobrushin, “Central limit theorems for non-stationary Markov chains. I,” Theory of Probability & Its Applications, vol. 1, pp. 65–80, 1956.

R. Dobrushin, “Central limit theorem for nonstationary Markov chains. II,” Theory of Probability & Its Applications, vol. 1, no. 4, pp. 329–383, 1956.

V. Statuljavicius, “Limit theorems for sums of random variables connected in Markov chains,” Lietuvos Matematikos Rinkinys, vol. 9, pp. 346–362, 635–672, Lietuvos Matematikos Rinkinys, vol. 10, pp. 583–592, 1969

T. A. Sarymsakov, “Inhomogeneous Markov chains,” Theory of Probability and Its Applications, vol. 6, no. 2, pp. 178–185, 1961.

S. Sethuraman and S. R. S. Varadhan, “A martingale proof of Dobrushin's theorem for non-homogeneous Markov chains,” Electronic Journal of Probability, vol. 10, article 36, pp. 1221–1235, 2005.

B. M. Brown, “Martingale central limit theorems,” Annals of Mathematical Statistics, vol. 42, pp. 59–66, 1971.