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

Abstract

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={𝑋_𝑛,𝑛≥0} be an arbitrary information source taking values on alphabet set 𝑆={1,2,…,𝑁} with the joint distribution 𝑃𝑋₀=𝑥₀,𝑋₁=𝑥₁,…,𝑋_𝑛=𝑥_𝑛=𝑝𝑥₀,𝑥₁,…,𝑥_𝑛,𝑥_𝑖∈𝑆, (1) for ∀0≤𝑖≤𝑛,𝑛≥0. If 𝑋={𝑋_𝑛,𝑛≥0} is an 𝑚th-order nonhomogeneous Markov information source, then, for 𝑛≥𝑚, 𝑃𝑋_𝑛=𝑥_𝑛∣𝑋₀=𝑥₀,𝑋₁=𝑥₁,…,𝑋_𝑛−1=𝑥_𝑛−1=𝑃𝑋_𝑛=𝑥_𝑛∣𝑋_𝑛−𝑚=𝑥_𝑛−𝑚,𝑋_{𝑛−𝑚+1}=𝑥_{𝑛−𝑚+1},…,𝑋_𝑛−1=𝑥_𝑛−1. (2) Denote 𝜇𝑖₀,𝑖₁,…,𝑖_𝑚−1=𝑃𝑋₀=𝑖₀,𝑋₁=𝑖₁,…,𝑋_𝑚−1=𝑖_𝑚−1,𝑝_𝑛𝑗∣𝑖₁,𝑖₂,…,𝑖_𝑚=𝑃𝑋_𝑛=𝑗∣𝑋_𝑛−𝑚=𝑖₁,…,𝑋_𝑛−1=𝑖_𝑚, (3)(4) where 𝜇(𝑖₀,𝑖₁,…,𝑖_𝑚−1) and 𝑝_𝑛(𝑗∣𝑖₁,𝑖₂,…,𝑖_𝑚) are called the m-dimensional initial distribution and the 𝑚th-order transition probabilities, respectively. Moreover, 𝑃_𝑛=𝑝_𝑛𝑗∣𝑖₁,𝑖₂,…,𝑖_𝑚 (5) are called the 𝑚th-order transition probability matrices. In this case, 𝑝𝑥₀,𝑥₁,…,𝑥_𝑛=𝜇𝑥₀,𝑥₁,…,𝑥_𝑚−1^𝑛_𝑘=𝑚𝑝_𝑘𝑥_𝑘∣𝑥_𝑘−𝑚,…,𝑥_𝑘−1. (6)

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 [1] 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 [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 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={𝑋_𝑛,𝑛≥0} 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 𝑋^𝑛_𝑚=𝑋_𝑚,𝑋_𝑚+1,…,𝑋_𝑛. (7) We also denote the realizations of 𝑋^𝑛_𝑚 by 𝑥^𝑛_𝑚. We denote the 𝑚th-order transition matrix at step 𝑘 by 𝑃_𝑘=𝑝_𝑘𝑗∣𝑖^𝑚₁,𝑗∈𝑆,𝑖^𝑚₁∈𝑆^𝑚, (8) where 𝑃_𝑘(𝑗∣𝑖^𝑚₁)=𝑃(𝑋_𝑘=𝑗∣𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁).

For an arbitrary stochastic square matrix 𝐴 whose elements are 𝐴_𝑖,𝑗, we will set the ergodic 𝛿-coefficient equal to 𝛿(𝐴)=sup_{𝑖,𝑗∈𝑆𝑘∈𝑆}𝐴_𝑖,𝑘−𝐴_𝑗,𝑘⁺, (9) where [𝑎]⁺=max{0,𝑎}. Now we extend this idea to the 𝑚th-order stochastic matrix 𝑄 whose elements are 𝑞(𝑖^𝑚₁,𝑗)=𝑞(𝑗∣𝑖^𝑚₁), and we will introduce the ergodic 𝛿-coefficient equal to 𝛿(𝑄)=sup_{𝑖𝑚1,𝑗𝑚1∈𝑆𝑚𝑘∈𝑆}𝑞𝑖^𝑚₁,𝑘−𝑞𝑗^𝑚₁,𝑘⁺. (10)

Now we define another stochastic matrix as follows: 𝑃=𝑝𝑗^𝑚₁∣𝑖^𝑚₁𝑖^𝑚₁,𝑗^𝑚₁∈𝑆^𝑚, (11) where 𝑝𝑗^𝑚₁∣𝑖^𝑚₁=𝑝𝑗_𝑚∣𝑖^𝑚₁,as𝑗_𝑣=𝑖_𝑣+1,𝑣=1,2,…,𝑚−1;0,otherwise. (12)𝑃 is called the m-dimensional stochastic matrix determined by the 𝑚th-order transition matrix.

Let 𝑆_𝑛(𝑖^𝑚₁)=𝑆_𝑛(𝑖₁,𝑖₂,…,𝑖_𝑚) be the number of (𝑖₁,𝑖₂,…,𝑖_𝑚) in the sequence of 𝑋₀^𝑚−1,𝑋^𝑚₁,…,𝑋^𝑛−1_𝑛−𝑚; that is, 𝑆_𝑛𝑖^𝑚₁=^𝑛_𝑘=𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁. (13)

Lemma 1 (see [1]). Let 𝑋={𝑋_𝑛,𝑛≥0} 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). 𝑆_𝑛(𝑖^𝑚₁) is defined as (13). Let 𝑃=(𝑝(𝑗∣𝑖^𝑚₁)) 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 lim_𝑛→∞1𝑛^𝑛_𝑘=𝑚𝑝_𝑘𝑗𝑖^𝑚₁−𝑝𝑗∣𝑖^𝑚₁=0,∀𝑗∈𝑆,𝑖^𝑚₁∈𝑆^𝑚. (14) Then one has lim_𝑛→∞𝑆_𝑛𝑖^𝑚₁𝑛=𝜋𝑖^𝑚₁,𝑎.𝑠. (15)

2. Statement of the Main Result

Let 𝑓(𝑥^𝑚₀) be any Borel function defined on product space 𝑆^𝑚+1. Denote 𝑊_𝑛=^𝑛_𝑘=𝑚𝐷_𝑘, (16) where 𝐷_𝑘=𝑓𝑋^𝑘_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚,𝐷_𝑚−1=0. (17) Obviously, {𝑊_𝑛,ℱ_𝑛,𝑛≥𝑚} is a martingale, so that {𝐷_𝑛,ℱ_𝑛,𝑛≥𝑚} is the associated martingale difference sequence. Denote 𝑆_𝑛=^𝑛_𝑘=𝑚𝑓𝑋^𝑘_𝑘−𝑚. (18)

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

Theorem 2. Let 𝑋={𝑋_𝑛,𝑛≥0} 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(𝑗∣𝑖^𝑚₁))_{𝑗∈𝑆,𝑖}^𝑚_1∈𝑆^𝑚 be an 𝑚-th order transition matrix. Let 𝑓 be any function defined on the state space 𝑆^𝑚+1 and let {𝑆_𝑛,𝑛≥0} be defined as (13). If (14) holds and the sequence of 𝛿-coefficients for the 𝑚th-order stochastic matrices {𝑃_𝑘,𝑘≥𝑚} satisfies that lim_𝑛→∞∑^𝑛_𝑘=𝑚𝛿𝑃_𝑘√𝑛=0, (19) then one has 𝑆_𝑛√𝑛𝜎^𝐷⇒𝑁(0,1), (20) where ⇒^𝐷 denotes the convergence in distribution and 𝜎²=_{𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚_1𝑗∈𝑆𝑓²𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁−_𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝(𝑗∣𝑖^𝑚₁)²>0. (21)

Remark 3. The sequence {𝑃_𝑘=(𝑝_𝑘(𝑗∣𝑖^𝑚₁)),𝑘≥𝑚} is said to converge in the Cesàro sense to constant matrix 𝑃=(𝑝(𝑗∣𝑖^𝑚₁)) if (14) holds.

3. Proof of Theorem 2

Let {Ω,ℱ,𝐏} be a probability space and let {𝑀_𝑛,𝑛=1,2,…} be a sequence of random variables which is defined on {Ω,ℱ,𝐏}. Let {ℱ_𝑛,𝑛=1,2,…} be an increasing sequence of 𝜎-fields of ℱ sets. Now let {𝑀_𝑛,ℱ_𝑛,𝑛=1,2,…} be a sequence of martingale, so that 𝐷₀=0,𝐷_𝑛=𝑀_𝑛−𝑀_𝑛−1,𝑛=1,2,…. (22) is a martingale difference. ℱ₀ is a trivial 𝜎 field. For 𝑛=1,2,…, denote 𝜎²_𝑛=𝐸𝐷²_𝑛∣ℱ_𝑛−1,𝑉²_𝑛=^𝑛_𝑗=1𝜎²_𝑗,𝑣²_𝑛=𝐸𝑉²_𝑛=𝐸𝑀²_𝑛. (23)

Lindeberg Condition. For ∀𝜖>0, lim_𝑛→∞∑^𝑛_𝑗=1𝐸𝐷²_𝑗𝐼𝐷_𝑗≥𝜖𝑣_𝑛𝑣²_𝑛=0, (24) 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 [15]). Suppose that the sequence of martingale {𝑀_𝑛,ℱ_𝑛,𝑛=1,2,…} satisfies the following condition: 𝑉²_𝑛𝑣²_𝑛^𝑝⟹1. (25) Moreover, if the Lindeberg condition holds, then one has 𝑀_𝑛𝑣_𝑛^𝐷⟹𝑁(0,1), (26) 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={𝑋_𝑛,𝑛≥0} 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 𝑆^𝑚+1. Suppose that the function 𝑓 satisfies condition (21). Let {𝑊_𝑛,𝑛≥0} be defined as (16). If (14) holds, then 𝑊_𝑛√𝑛𝜎^𝐷⇒𝑁(0,1), (27) where ⇒^𝐷 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 𝑉²_𝑛𝑛=1𝑛^𝑛_𝑘=𝑚𝐸𝐷²_𝑘∣ℱ_𝑘−1=1𝑛^𝑛_𝑘=𝑚𝐸𝑓²𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚²∶=𝐼₁(𝑛)−𝐼₂(𝑛), (28) where 𝐼₁(𝑛)=1𝑛^𝑛_𝑘=𝑚𝐸𝑓²𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚=1𝑛^𝑛_{𝑘=𝑚𝑗∈𝑆𝑖𝑚1∈𝑆𝑚}𝑓²𝑖^𝑚₁,𝑗𝑝_𝑘𝑗∣𝑖^𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁=_{𝑗∈𝑆𝑖𝑚1∈𝑆𝑚}𝑓²𝑖^𝑚₁1,𝑗𝑛^𝑛_𝑘=𝑚𝑝_𝑘𝑗∣𝑖^𝑚₁𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁,𝐼₂(𝑛)=1𝑛^𝑛_𝑘=𝑚𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚²=1𝑛^𝑛_{𝑘=𝑚𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝_𝑘𝑗∣𝑖^𝑚₁²×𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁. (29)(30) Noting that, on the one hand, 1𝑛^𝑛_𝑘=𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁𝑝_𝑘𝑗∣𝑖^𝑚₁−𝑝𝑗∣𝑖^𝑚₁≤1𝑛^𝑛_𝑘=𝑚𝑝_𝑘𝑗∣𝑖^𝑚₁−𝑝𝑗∣𝑖^𝑚₁ (31) which tends to zero as 𝑛 tends to infinity by using (14). Thus we have lim_𝑛→∞1𝑛^𝑛_𝑘=𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁𝑝_𝑘𝑗∣𝑖^𝑚₁=lim_𝑛→∞1𝑛^𝑛_𝑘=𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁𝑝𝑗∣𝑖^𝑚₁=lim_𝑛→∞1𝑛𝑆_𝑛𝑖^𝑚₁𝑝𝑗∣𝑖^𝑚₁=𝜋𝑖^𝑚₁𝑝𝑗∣𝑖^𝑚₁a.s., (32) where the third equation holds because of (15). Combining (29) and (32), we get lim_𝑛→∞𝐼₁(𝑛)=_{𝑗∈𝑆𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚₁𝑝𝑗∣𝑖^𝑚₁𝑓²𝑖^𝑚₁,𝑗a.s. (33) 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 𝐼₂(𝑛)−1𝑛^𝑛_{𝑘=𝑚𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁≤1𝑛^𝑛_{𝑘=𝑚𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝_𝑘𝑗∣𝑖^𝑚₁−𝑝𝑗∣𝑖^𝑚₁×_𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝_𝑘𝑗∣𝑖^𝑚₁+𝑝𝑗∣𝑖^𝑚₁≤2max_{𝑖𝑚1∈𝑆𝑚,𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗²×_{𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}∑^𝑛_𝑘=𝑚𝑝_𝑘𝑗∣𝑖^𝑚₁−𝑝𝑗∣𝑖^𝑚₁𝑛⟶0.as𝑛⟶∞. (34)
Thus by Lemma 1, we easily arrive at lim_𝑛→∞𝐼₂(𝑛)=_{𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²1𝑛^𝑛_𝑘=𝑚𝐼𝑋^𝑘−1_𝑘−𝑚=𝑖^𝑚₁=_{𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²𝑆_𝑛𝑖^𝑚₁𝑛=_{𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚_1𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²a.s. (35)
Combining (28), (33), and (35), we arrive at lim_𝑛→∞𝑉²_𝑛𝑛=_{𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚_1𝑗∈𝑆𝑓²𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁−_𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²a.s., (36) which implies that lim_𝑛→∞𝑉²_𝑛𝑛=_{𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚_1𝑗∈𝑆𝑓²𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁−_𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²inprobability. (37) Note that 𝑉²_𝑛𝑛≤max_{𝑚≤𝑘≤𝑛}𝐸𝐷²_𝑘∣𝑋^𝑘−1_𝑘−𝑚=max_{𝑚≤𝑘≤𝑛}𝐸𝑓²𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚²≤max_{𝑖𝑚1∈𝑆𝑚,𝑗∈𝑆}𝑓²𝑖^𝑚₁,𝑗. (38) Since 𝑆 is a finite set, then the random sequence {𝑉²_𝑛/𝑛,𝑛≥1} is uniformly integrable. Combining above two facts, we arrive at lim_𝑛→∞𝐸𝑉²_𝑛𝑛=_{𝑖𝑚1∈𝑆𝑚}𝜋𝑖^𝑚_1𝑗∈𝑆𝑓²𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁−_𝑗∈𝑆𝑓𝑖^𝑚₁,𝑗𝑝𝑗∣𝑖^𝑚₁²=𝜎²>0. (39)
It follows that 𝑉²_𝑛𝑣²_𝑛^𝑝⇒1, (40) where 𝑣²_𝑛=𝐸[𝑉²_𝑛]=𝐸[𝑊²_𝑛]. On the other hand, similar to the analysis of inequality (38), we also have that {𝐷²_𝑛=[𝑓(𝑋^𝑘_𝑘−𝑚)−𝐸[𝑓(𝑋^𝑘_𝑘−𝑚)∣𝑋^𝑘−1_𝑘−𝑚]]²} is uniformly integrable, so that lim_𝑛→∞∑^𝑛_𝑗=𝑚𝐄𝐷²_𝑗𝐼𝐷_𝑗≥𝜖√𝑛𝑛=0, (41) 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 𝑆_𝑛−𝐸𝑆_𝑛=𝑊_𝑛+^𝑛_𝑘=𝑚𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚. (42)
Denote 𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁,𝑋_𝑘=𝑗=𝑃_𝑘𝑠^𝑚₁,𝑗. (43) and 𝑀=sup_{𝑠𝑚1∈𝑆}^𝑚_,𝑗∈𝑆𝑓(𝑠^𝑚₁,𝑗). Let us come to estimate the upper bound of |𝐸[𝑓(𝑋^𝑘_𝑘−𝑚)∣𝑋_𝑘−1]−𝐸[𝑓(𝑋^𝑘_𝑘−𝑚)]|. In fact, it follows from the C-K formula 𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚=_𝑗∈𝑆𝑓𝑋^𝑘−1_𝑘−𝑚,𝑗𝑃_𝑘𝑗∣𝑋^𝑘−1_𝑘−𝑚−_{𝑠𝑚1∈𝑆𝑚,𝑗∈𝑆}𝑓𝑠^𝑚₁,𝑗𝑃_𝑘𝑠^𝑚₁,𝑗≤sup_{𝑖𝑚1𝑗∈𝑆}𝑓𝑖^𝑚₁,𝑗𝑝_𝑘𝑗∣𝑖^𝑚₁−_𝑠𝑚1𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁𝑝_𝑘𝑗∣𝑠^𝑚₁≤𝑀sup_{𝑖𝑚1𝑗}𝑝_𝑘𝑗∣𝑖^𝑚₁−_𝑠𝑚1𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁𝑝_𝑘𝑗∣𝑠^𝑚₁=𝑀sup_{𝑖𝑚1𝑗𝑠𝑚1}𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁𝑝_𝑘𝑗∣𝑖^𝑚₁−_𝑠𝑚1𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁𝑝_𝑘𝑗∣𝑠^𝑚₁≤𝑀_𝑠𝑚1𝑃𝑋^𝑘−1_𝑘−𝑚=𝑠^𝑚₁×sup_{𝑖𝑚1∈𝑆𝑚𝑗∈𝑆}𝑝_𝑘𝑗∣𝑖^𝑚₁−𝑝_𝑘𝑗∣𝑠^𝑚₁≤𝑀sup_{𝑙𝑚1,𝑘𝑚1𝑗∈𝑆}𝑝_𝑘𝑗∣𝑙^𝑚₁−𝑝_𝑘𝑗∣𝑘^𝑚₁=2𝑀𝛿𝑃_𝑘, (44) where 𝛿(𝑃_𝑘)=sup_𝑙^𝑚_1,𝑘^𝑚₁∑_𝑗∈𝑆[𝑝_𝑘(𝑗∣𝑙^𝑚₁)−𝑝_𝑘(𝑗∣𝑘^𝑚₁)]⁺=(1/2)sup_𝑙^𝑚_1,𝑘^𝑚₁∑_𝑗∈𝑆|𝑝_𝑘(𝑗∣𝑙^𝑚₁)−𝑝_𝑘(𝑗∣𝑘^𝑚₁)|. By using condition (19), we get lim_𝑛→∞∑^𝑛_𝑘=𝑚𝐸𝑓𝑋^𝑘_𝑘−𝑚∣𝑋^𝑘−1_𝑘−𝑚−𝐸𝑓𝑋^𝑘_𝑘−𝑚√𝑛=0. (45) 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.