JPS Journal of Probability and Statistics 1687-9538 1687-952X Hindawi Publishing Corporation 645151 10.1155/2013/645151 645151 Research Article The Central Limit Theorem for mth-Order Nonhomogeneous Markov Information Source Huang Huilin Chow Shein-chung College of Mathematics and Information Science Wenzhou University Zhejiang 325035 China wzu.edu.cn 2013 11 12 2013 2013 22 08 2013 17 10 2013 2013 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 mth-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 X={Xn,n0} be an arbitrary information source taking values on alphabet set S={1,2,,N} with the joint distribution (1)P(X0=x0,X1=x1,,Xn=xn)=p(x0,x1,,xn),xiS, for 0in,n0. If X={Xn,n0} is an mth-order nonhomogeneous Markov information source, then, for nm, (2)P(Xn=xnX0=x0,X1=x1,,Xn-1=xn-1)=P(Xn=xnXn-m=xn-m,hhhhhhhhhhhhhhXn-m+1=xn-m+1,,Xn-1=xn-1). Denote (3)μ(i0,i1,,im-1)=P(X0=i0,X1=i1,,Xm-1=im-1),(4)pn(ji1,i2,,im)=P(Xn=jXn-m=i1,,Xn-1=im), where μ(i0,i1,,im-1) and pn(ji1,i2,,im) are called the m-dimensional initial distribution and the mth-order transition probabilities, respectively. Moreover, (5)Pn=(pn(ji1,i2,,im)) are called the mth-order transition probability matrices. In this case, (6)p(x0,x1,,xn)=μ(x0,x1,,xm-1)k=mnpk(xkxk-m,,xk-1).

There are many of practical information sources, such as language and image information, which are often mth-order Markov information sources and always nonhomogeneous. So it is very important to study the limit properties for the mth-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 mth-order nonhomogeneous Markov information sources. But the problem about the central limit theorem for the mth-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 . 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 1th-order nonhomogeneous Markov chain. In this paper, we come to study the central limit theorem for mth-order nonhomogeneous Markov information sources in Cesàro sense.

Let X={Xn,n0} be an mth-order nonhomogeneous Markov information source which is taking values in state space S={1,2,,N} with initial distribution of (3) and mth order transition probability matrices (5). Denote (7)Xmn={Xm,Xm+1,,Xn}. We also denote the realizations of Xmn by xmn. We denote the mth-order transition matrix at step k by (8)Pk=(pk(ji1m)),jS,i1mSm, where Pk(ji1m)=P(Xk=jXk-mk-1=i1m).

For an arbitrary stochastic square matrix A whose elements are Ai,j, we will set the ergodic δ-coefficient equal to (9)δ(A)=supi,jSkS[Ai,k-Aj,k]+, where [a]+=max{0,a}. Now we extend this idea to the mth-order stochastic matrix Q whose elements are q(i1m,j)=q(ji1m), and we will introduce the ergodic δ-coefficient equal to (10)δ(Q)=supi1m,j1mSmkS[q(i1m,k)-q(j1m,k)]+.

Now we define another stochastic matrix as follows: (11)P-=(p-(j1mi1m))i1m,j1mSm, where (12)p-(j1mi1m)={p(jmi1m),asjv=iv+1,v=1,2,,m-1;0,otherwise.P- is called the m-dimensional stochastic matrix determined by the mth-order transition matrix.

Let Sn(i1m)=Sn(i1,i2,,im) be the number of (i1,i2,,im) in the sequence of X0m-1,X1m,,Xn-mn-1; that is, (13)Sn(i1m)=k=mnI{Xk-mk-1=i1m}.

Lemma 1 (see [<xref ref-type="bibr" rid="B20">1</xref>]).

Let X={Xn,n0} be an mth-order nonhomogeneous Markov information source which is taking values in state space S={1,2,,N} with initial distribution of (3) and mth-order transition probability matrices (5). Sn(i1m) is defined as (13). Let P=(p(ji1m)) be another m-order transition matrix, and let P- be the m-dimensional stochastic matrix determined by the mth-order transition matrix P, that is, π=πP-. Suppose that (14)limn1nk=mn|pk(ji1m)-p(ji1m)|=0,jS,i1mSm. Then one has (15)limnSn(i1m)n=π(i1m),a.s.

2. Statement of the Main Result

Let f(x0m) be any Borel function defined on product space Sm+1. Denote (16)Wn=k=mnDk, where (17)Dk=f(Xk-mk)-E[f(Xk-mk)Xk-mk-1],Dm-1=0. Obviously, {Wn,n,nm} is a martingale, so that {Dn,n,nm} is the associated martingale difference sequence. Denote (18)Sn=k=mnf(Xk-mk).

Our main result is describing conditions on X and f under in which the central limit theorem holds for the stochastic sequence Sn.

Theorem 2.

Let X={Xn,n0} be an mth-order nonhomogeneous Markov information source which is taking values in state space S={1,2,,N} with initial distribution of (3) and mth-order transition probability matrices (5). Let P=(P(ji1m))jS,i1mSm be an m-th order transition matrix. Let f be any function defined on the state space Sm+1 and let {Sn,n0} be defined as (13). If (14) holds and the sequence of δ-coefficients for the mth-order stochastic matrices {Pk,km} satisfies that (19)limnk=mnδ(Pk)n=0, then one has (20)SnnσDN(0,1), where D denotes the convergence in distribution and (21)σ2=i1mSmπ(i1m){[jSf(i1m,j)p(ji1m)]2jSf2(i1m,j)p(ji1m)=i1mSmπ(i1m)h-[jSf(i1m,j)p(ji1m)]2}>0.

Remark 3.

The sequence {Pk=(pk(ji1m)),km} is said to converge in the Cesàro sense to constant matrix P=(p(ji1m)) if (14) holds.

3. Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref>

Let {Ω,,P} be a probability space and let {Mn,n=1,2,} be a sequence of random variables which is defined on {Ω,,P}. Let {n,n=1,2,} be an increasing sequence of σ-fields of sets. Now let {Mn,n,n=1,2,} be a sequence of martingale, so that (22)D0=0,Dn=Mn-Mn-1,n=1,2,. is a martingale difference. 0 is a trivial σ field. For n=1,2,, denote (23)σn2=E(Dn2n-1),Vn2=j=1nσj2,vn2=E(Vn2)=E(Mn2).

Lindeberg Condition. For ϵ>0, (24)limnj=1nEDj2I{|Dj|ϵvn}vn2=0, where I{·} denotes the index function.

In our proof, we will use the central limit theorem of martingale sequences as the technical tool.

Lemma 4 (see [<xref ref-type="bibr" rid="B1">15</xref>]).

Suppose that the sequence of martingale {Mn,n,n=1,2,} satisfies the following condition: (25)Vn2vn2p1. Moreover, if the Lindeberg condition holds, then one has (26)MnvnDN(0,1), where p and D 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 X={Xn,n0} be an m-order nonhomogeneous Markov information source which is taking values in state space S={1,2,,N} with initial distribution of (3) and mth-order transition probability matrices (5). Let f be any function defined on the state space Sm+1. Suppose that the function f satisfies condition (21). Let {Wn,n0} be defined as (16). If (14) holds, then (27)WnnσDN(0,1), where D denotes the convergence in distribution.

Proof of Theorem <xref ref-type="statement" rid="thm3.2">5</xref>.

Noting that by using the property of the conditional expectation and Markov property, it follows from (17) that (28)Vn2n=1nk=mnE[Dk2k-1]=1nk=mn{(E[f(Xk-mk)Xk-mk-1])2E[f2(Xk-mk)Xk-mk-1]=1nk=mnh-(E[f(Xk-mk)Xk-mk-1])2}:=I1(n)-I2(n), where (29)I1(n)=1nk=mnE[f2(Xk-mk)Xk-mk-1]=1nk=mnjSi1mSmf2(i1m,j)pk(jim)I{Xk-mk-1=i1m}=jSi1mSmf2(i1m,j)1nk=mnpk(ji1m)I{Xk-mk-1=i1m},(30)I2(n)=1nk=mn(E[f(Xk-mk)Xk-mk-1])2=1nk=mni1mSm[jSf(i1m,j)pk(ji1m)]2×I{Xk-mk-1=i1m}. Noting that, on the one hand, (31)|1nk=mnI{Xk-mk-1=i1m}[pk(ji1m)-p(ji1m)]|1nk=mn|pk(ji1m)-p(ji1m)| which tends to zero as n tends to infinity by using (14). Thus we have (32)limn1nk=mnI{Xk-mk-1=i1m}pk(ji1m)=limn1nk=mnI{Xk-mk-1=i1m}p(ji1m)=limn1nSn(i1m)p(ji1m)=π(i1m)p(ji1m)a.s., where the third equation holds because of (15). Combining (29) and (32), we get (33)limnI1(n)=jSi1mSmπ(i1m)p(ji1m)f2(i1m,j)a.s. On the other hand, let us come to compute the limit of I2(n) as n tends to infinity. By using (14) again, we have (34)|I2(n)-1nk=mni1mSm[jSf(i1m,j)p(ji1m)]2I{Xk-mk-1=i1m}|1nk=mni1mSm[jSf(i1m,j)|pk(ji1m)-p(ji1m)|]hhhhhhh×[jSf(i1m,j)(pk(ji1m)+p(ji1m))]2(maxi1mSm,jSf(i1m,j))2×i1mSmjSk=mn|pk(ji1m)-p(ji1m)|n0.  hhhhhhhhhhhhhhhhhhhhhhhhhasn  .

Thus by Lemma 1, we easily arrive at (35)limnI2(n)=i1mSm[jSf(i1m,j)p(ji1m)]21nk=mnI{Xk-mk-1=i1m}=i1mSm[jSf(i1m,j)p(ji1m)]2Sn(i1m)n=i1mSmπ(i1m)[jSf(i1m,j)p(ji1m)]2a.s.

Combining (28), (33), and (35), we arrive at (36)limnVn2n=i1mSmπ(i1m){[jSf(i1m,j)p(ji1m)]2jSf2(i1m,j)p(ji1m)=i1mSmπ(i1m)h-[jSf(i1m,j)p(ji1m)]2}a.s., which implies that (37)limnVn2n=i1mSmπ(i1m){[jSf(i1m,j)p(ji1m)]2jSf2(i1m,j)p(ji1m)=i1mSmπ(i1m)h-[jSf(i1m,j)p(ji1m)]2}hhhhhhhhhhhhhhhhhhhhhhin  probability. Note that (38)Vn2nmaxmknE[Dk2Xk-mk-1]=maxmkn{(E[f(Xk-mk)Xk-mk-1])2E[f2(Xk-mk)Xk-mk-1]=maxmknh-(E[f(Xk-mk)Xk-mk-1])2}maxi1mSm,jSf2(i1m,j). Since S is a finite set, then the random sequence {Vn2/n,n1} is uniformly integrable. Combining above two facts, we arrive at (39)limnE[Vn2]n=i1mSmπ(i1m){jSf2(i1m,j)p(ji1m)=i1mSmπ(i1m)hh-[jSf(i1m,j)p(ji1m)]2}=σ2>0.

It follows that (40)Vn2vn2p1, where vn2=E[Vn2]=E[Wn2]. On the other hand, similar to the analysis of inequality (38), we also have that {Dn2=[f(Xk-mk)-E[f(Xk-mk)Xk-mk-1]]2} is uniformly integrable, so that (41)limnj=mnEDj2I(|Dj|ϵn)n=0, 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 <xref ref-type="statement" rid="thm2.1">2</xref>.

Note that (42)Sn-E[Sn]=Wn+k=mn[E[f(Xk-mk)Xk-mk-1]-E[f(Xk-mk)]].

Denote (43)P(Xk-mk-1=s1m,Xk=j)=Pk(s1m,j). and M=sups1mSm,jSf(s1m,j). Let us come to estimate the upper bound of |E[f(Xk-mk)Xk-1]-E[f(Xk-mk)]|. In fact, it follows from the C-K formula (44)|E[f(Xk-mk)Xk-mk-1]-E[f(Xk-mk)]|=|jSf(Xk-mk-1,j)Pk(jXk-mk-1)h-s1mSm,jSf(s1m,j)Pk(s1m,j)|supi1m|jSf(i1m,j)[s1mpk(ji1m)hhhjSf(i1m,j)hh-s1mP(Xk-mk-1=s1m)pk(js1m)]|Msupi1mj|pk(ji1m)-s1mP(Xk-mk-1=s1m)pk(js1m)|=Msupi1mj|s1mP(Xk-mk-1=s1m)pk(ji1m)hhhhhhhh-s1mP(Xk-mk-1=s1m)pk(js1m)|Ms1mP(Xk-mk-1=s1m)hhh×supi1mSmjS|pk(ji1m)-pk(js1m)|Msupl1m,k1mjS|pk(jl1m)-pk(jk1m)|=2Mδ(Pk), where δ(Pk)=supl1m,k1mjS[pk(jl1m)-pk(jk1m)]+=(1/2)supl1m,k1mjS|pk(jl1m)-pk(jk1m)|. By using condition (19), we get (45)limnk=mn[E[f(Xk-mk)Xk-mk-1]-E[f(Xk-mk)]]n=0. 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.

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