Some Limit Properties of Random Transition Probability for Second Order Nonhomogeneous Markov Chains Indexed by a Tree

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Volume 2009, Article ID 503203,10pages doi:10.1155/2009/503203

Research Article

Some Limit Properties of Random Transition

Probability for Second-Order Nonhomogeneous

Markov Chains Indexed by a Tree

Zhiyan Shi and Weiguo Yang

Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Correspondence should be addressed to Zhiyan Shi,[email protected]

Received 1 September 2009; Accepted 24 November 2009

Recommended by Andrei Volodin

We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.

Copyrightq2009 Z. Shi and W. Yang. 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.

1. Introduction

A tree is a graph G {T, E} which is connected and contains no circuits. Given any two verticesσ, tσ /t∈T, letσtbe the unique path connectingσandt. Define the graph distance

dσ, tto be the number of edges contained in the pathσt.

LetTobe an arbitrary infinite tree that is partially finitei.e., it has infinite vertices, and

each vertex connects with finite verticesand has a rooto. Meanwhile, we consider another kind of double root tree T; that is, it is formed with the root o of To connecting with an

arbitrary point denoted by the root−1. For a better explanation of the double root tree T, we take Cayley treeTC,N for example. It is a special case of the treeTo, the rootoof Cayley

tree hasNneighbors, and all the other vertices of it haveN1 neighbors each. The double root treeT_C,N seeFigure 1is formed with rootoof treeTC,N connecting with another root

−1.

Letσ,tbe vertices of the double root treeT. Writet≤σσ, t / −1iftis on the unique path connectingotoσ, and|σ|for the number of edges on this path. For any two verticesσ,

tσ, t / −1of the treeT, denote byσ∧tthe vertex farthest fromosatisfyingσ∧t ≤σand

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Level 3 Level 2 Level 1

Level 0 Level−1

t

1t

2t

Rooto

[image:2.600.195.404.97.252.2]

Root−1

Figure 1:Double root treeT_C, ₂.

The set of all vertices with distance nfrom rooto is called thenth generation ofT, which is denoted byLn. We say thatLnis the set of all vertices on levelnand especially root

−1 is on the−1st level on treeT. We denote byTn _{the subtree of the tree}_T _{containing the} vertices from level−1 the root−1to levelnand denote byTon the subtree of the treeTo

containing the vertices from level 0the roototo leveln. Lett/o,−1be a vertex of the tree

T. We denote the first predecessor oftby 1t, the second predecessor oftby 2t, and denote by

ntthenth predecessor oft. LetXA {Xt, t∈A}, and letxAbe a realization ofXAand denote

by|A|the number of vertices ofA.

Definition 1.1. LetG{1,2, . . . , N}andPz|y, xbe nonnegative functions onG3_{. Let}

P Pz|y, x, Pz|y, x≥0, x, y, z∈G. 1.1

If

z∈G

Pz|y, x1, 1.2

thenPis called a second-order transition matrix.

Definition 1.2. LetT be double root tree and letG {1,2, . . . , N}be a finite state space, and let{Xt, t∈T}be a collection ofG-valued random variables defined on the probability space

Ω,F, P. Let

P px, y, x, y∈G 1.3

be a distribution onG2, and

Pt

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be a collection of second-order transition matrices. For any vertex tt /o,−1, if

PXtz|X1ty, X2t x, Xσ forσ∧t≤1t

PXtz|X1ty, X2tx

Pt

z|y, x ∀x, y, z∈G, PX−1x, Xoy

px, y, x, y∈G,

1.5

then{Xt, t∈T}is called aG-value second-order nonhomogeneous Markov chain indexed by

a treeT with the initial distribution1.3and second-order transition matrices1.4, or called aT-indexed second-order nonhomogeneous Markov chain.

Remark 1.3. Benjamini and Peres 1 have given the definition of the tree-indexed homogeneous Markov chains. Here we improve their definition and give the definition of the tree-indexed second-order nonhomogeneous Markov chains in a similar way. We also give the following definitionDefinition 2.3of tree-indexed nonhomogeneous Markov chains.

There have been some works on limit theorems for tree-indexed stochastic processes. Benjamini and Peres1have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye 2 have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Ye and Berger see 3, 4, by using Pemantle’s result 5 and a combinatorial approach, have studied the Shannon-McMillan theorem with convergence in probability for a PPG-invariant and ergodic random field on a homogeneous tree. Yang and Liu6have studied a strong law of large numbers for the frequency of occurrence of states for Markov chains field on a homogeneous treea particular case of tree-indexed Markov chains field and PPG-invariant random fields. Yang see7 has studied the strong law of large numbers for frequency of occurrence of state and Shannon-McMillan theorem for homogeneous Markov chains indexed by a homogeneous tree. Recently, Yangsee8has studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang see9 have also studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree.

LetPtxt | x1t, x2t PtXt xt | X1t x1t, X2t x2t. ThenPtXt |X1t, X2tis called the random transition probability of a T-indexed second-order nonhomogeneous Markov chain. Liu 10has studied a strong limit theorem for the harmonic mean of the random transition probability of finite nonhomogeneous Markov chains. In this paper, we study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the results of10,11.

2. Main Results

Lemma 2.1. Let{Xt, t∈T}be aT-indexed second-order nonhomogeneous Markov chain with state

spaceGdefined as inDefinition 1.2, and let{gtx, y, z, t∈T}be a collection of functions defined on

G3_{. Let}_L

−1{−1},L0{o}, andFnσXT

n . Set

tnλ, ω

eλt∈Tn\{o}{−1}gtX2t,X1t,Xt

t∈Tn_\{_o_}{−₁_}E

eλgtX2t,X1t,Xt|_X 1t, X2t

, 2.1

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Proof. Obviously, whenn≥1, we have

P xTnP XTn xTnPX−1x−1, Xoxo

t∈Tn_\{_o_}{−₁_}

Ptxt|x1t, x2t. 2.2

Hence

P XLn _xLn |_XTn−1_xTn−1

P x

Tn

PxTn−1

t∈Ln

Ptxt|x1t, x2t. 2.3

Then

Eeλt∈LngtX2t,X1t,Xt| F

n−1

xLn_∈_GLn

eλt∈LngtX2t,X1t,xt_P _XLn _xLn |_XTn−1

xLn_∈_GLn

eλt∈LngtX2t,X1t,xt

t∈Ln

Ptxt|X1t, X2t

t∈Ln

xt∈G

eλgtX2t,X1t,xt_P

txt|X1t, X2t

t∈Ln

EeλgtX2t,X1t,Xt|_X 1t, X2t

a.e.

2.4

On the other hand, we also have

tnλ, ω tn−1λ, ω e

λt∈LngtX2t,X1t,Xt

t∈LnE

eλgtX2t,X1t,Xt|_X 1t, X2t

. 2.5

Combining2.4and2.5, we arrive at

Etnλ, ω| Fn−1 tn−1λ, ω a.e., 2.6

Thus the lemma is proved.

Theorem 2.2. Let{Xt, t∈T}be aT-indexed second-order nonhomogeneous Markov chain with state

spaceG defined as inDefinition 1.2, and its initial distribution and probability transition collection satisfying

PX₋1x−1, Xoxo P

x, y>0, ∀x, y∈G, Pt

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respectively. Let

btmin

Pt

z|y, x, x, y, z∈G, t∈T\ {o}{−1}. 2.8

If there existsa>0such that

lim sup

n→ ∞

1 _Tn

t∈Tn_\{_o_}{−₁_}

ea/bt_{M <}∞_, _2.9

then the harmonic mean of the random conditional probability{PtXt|X1t, X2t, t∈Tn\ {o}{−1}} converges to1/Na.e., that is,

lim

n→ ∞

Tn

t∈Tn_\{_o_}{−₁_}PtXt|X1t, X2t −1

1

N a.e. 2.10

Proof. Letgtx, y, z Ptz|y, x−1inLemma 2.1. Then it follows fromLemma 2.1that

tnλ, ω

eλt∈Tn\{o}{−1}PtXt|X1t,X2t−

1

t∈Tn_\{_o_}{−₁_}E

eλPtXt|X1t,X2t−1 |X₁ t, X2t

2.11

is a nonnegative martingale. According to Doob martingale convergence theorem, we have

lim

n→ ∞tnλ, ω tλ, ω<∞ a.e. 2.12

Thus

lim sup

n→ ∞

1

_Tnlntnλ, ω≤0 a.e. 2.13

It follows from2.11and2.13that

lim sup

n→ ∞

1 _Tn

⎧ ⎨ ⎩λ

t∈Tn_\{_o_}{−₁_}

PtXt|X1t, X2t −1

−

t∈Tn_\{_o_}{−₁_}

lnEeλPtXt|X1t,X2t−1 |X₁ t, X2t

⎫⎬

⎭≤0 a.e.

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By2.14and the inequalities lnx≤x−1x >0, and 0≤ex₋₁₋_x_≤_x2_/₂_e|x|_{, we have}

lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

λPtXt|X1t, X2t

−1₋_λN

≤lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

lnEeλPtXt|X1t,X2t−

1

|X1t, X2t

−λN

≤lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

EeλPtXt|X1t,X2t−

1

|X1t, X2t

−1−λN

lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

xt∈G

Ptxt|X1t, X2t

eλPtxt|X1t,X2t−

1

−1−λPtxt|X1t, X2t −1

≤ λ2

2lim sup_n_{→ ∞} 1 Tn

t∈Tn_\{_o_}{−₁_}

xt∈G

Ptxt|X1t, X2t

−1_e|λ|Ptxt|X1t,X2t−1

≤ λ2

2lim sup_n_{→ ∞} 1 _Tn

t∈Tn_\{_o_}{−₁_}

xt∈G 1

bt

e|λ|/bt

≤ λ2N

2 lim supn→ ∞

1 _Tn

t∈Tn_\{_o_}{−₁_} 1

bt

e|λ|/bt _a_._e_.

2.15

It is easy to see that

max 0<λ<1{xλ

x_{, x >}₀_}₋e−1

lnλ. 2.16

Let 0< λ < a, by2.15,2.16,2.8, and2.9we have

lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

PtXt|X1t, X2t −1₋_N

≤ λN

2 lim sup_n_{→ ∞} 1 Tn

t∈Tn_\{_o_}{−₁_} 1

bt

eλ/bt

λN

2 lim sup_n_{→ ∞} 1 Tn

t∈Tn_\{_o_}{−₁_} 1

bt

eλ

ea

1/bt ea/bt

≤ λN

2a−λelim supn→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

ea/bt

λN

2a−λeM,

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Lettingλ → 0, by2.17, we have

lim sup

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

PtXt|X1t, X2t

−1₋_N_≤_{0 a}_._e_. _2.18

Let−a < λ <0, by2.15,2.8, and2.9we have

lim inf

n→ ∞

1 _Tn

t∈Tn_\{_o_}{−₁_}

PtXt|X1t, X2t −1₋_N

≥ λN

2 lim supn→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_} 1

bt

e−λ/bt

λN

2 lim supn→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_} 1

bt

e−λ

ea

1/bt ea/bt

≥ λN

2aλelim sup_n_{→ ∞}

1 _Tn

t∈Tn_\{_o_}{−₁_}

ea/bt

λN

2aλeM.

2.19

Lettingλ → 0−, by2.19, we have

lim inf

n→ ∞

1 Tn

t∈Tn_\{_o_}{−₁_}

PtXt|X1t, X2t

−1₋_N_≥_{0 a}_._e_. _2.20

Combining2.18and2.20, we obtain2.10directly.

From the definition above, we know that the diﬀerence between To and T lies in

whether the rootois connected with another root−1. In the following, we will investigate some properties of the harmonic mean of the transition probability of nonhomogeneous Markov chains on the treeTo. First, we give the definition of nonhomogeneous Markov chains

on the treeTo.

Definition 2.3. LetTobe an arbitrary tree that is partly finite, letG {1,2, . . . , N}be a finite

state space, and let{Xt, t∈To}be a collection ofG-valued random variables defined on the

probability spaceΩ,F, P. Let

Ppx, x∈G 2.21

be a distribution onG, and

Pt

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be a collection of transition matrices. For any vertextt /o, if

PXty|X1tx, Xσ forσ∧t≤1t

PXty|X1tx

Pt

y|x, ∀x, y∈G,

PXox px, x∈G,

2.23

then {Xt, t ∈ To} is called a G-value nonhomogeneous Markov chain indexed by a tree

To with the initial distribution2.21and transition matrices 2.22, or called a To-indexed

nonhomogeneous Markov chain.

Let Ptxt | x1t PtXt xt | X1t x1t. Then PtXt | X1tis called the random transition probability of aTo-indexed nonhomogeneous Markov chain. Since a Markov chain

is a special case of a second-order Markov chain, we may regard the nonhomogeneous Markov chain onToto be a special case of the second-order nonhomogeneous Markov chain

onTwhen we do not take the diﬀerence ofToandTon the root−1 into consideration. Thus for

the nonhomogeneous Markov chain on the treeTo, we can get the results similar toLemma 2.1

andTheorem 2.2.

Lemma 2.4. Let{Xt, t∈To}be aTo-indexed second-order nonhomogeneous Markov chain with state

spaceGdefined as inDefinition 2.3, and let{gtx, y, t ∈To}be a collection of functions defined on

G2_{. Let}_L

0{o}andFnσXT

n

o _{. Set}

tnλ, ω

eλ

t∈T_on\{o}gtX1t,Xt

t∈Ton\{o}E

eλgtX1t,Xt|_X 1t

, 2.24

whereλis a real number. Then{tnλ, ω,Fn, n≥1}is a nonnegative martingale.

Theorem 2.5. Let{Xt, t ∈ To}be aTo-indexed nonhomogeneous Markov chain with state spaceG

defined as inDefinition 2.3, and its initial distribution and probability transition collection satisfying

PXox px>0, ∀x∈G,

Pt

y|x>0, ∀x, y∈G, t∈To\ {o},

2.25

respectively. Let

btmin

Pt

y|x, x, y∈G, t∈To\ {o}. 2.26

lim sup

n→ ∞

1 Ton

t∈Ton\{o}

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then the harmonic mean of the random conditional probability {PtXt | X1t, t ∈ T

n

o \ {o}}

converges to1/Na.e., that is

lim

n→ ∞

Ton

t∈Ton\{o}PtXt|X1t −1

1

N a.e. 2.28

If the successor of each vertex of the tree To has only one vertex, then the

nonhomogeneous Markov chains on the treeTodegenerate into the general nonhomogeneous

Markov chains. Thus we obtain the results in10,11.

Corollary 2.6see10,11. Let{Xn, n≥0}be a nonhomogeneous Markov chain with state space

G, and its initial distribution and probability transition sequence satisfying

pi>0, i∈G, Pk

i, j>0, i, j∈G, k1,2, . . . ,

2.29

respectively. Let

akmin

Pk

i, j, i, j∈G, k1,2, . . . . 2.30

lim sup

n→ ∞

1

n

k1

ea/ak _{M <}∞_, _2.31

then

lim

n→ ∞

n

k1PkXk|Xk−1−1

1

N a.e. 2.32

Proof. When the successor of each vertex of the tree To has only one vertex, the

nonhomogeneous Markov chains on the treeTodegenerate into the general nonhomogeneous

Markov chains, the corollary follows directly fromTheorem 2.5.

Acknowledgments

This work is supported by the National Natural Science Foundation of China10571076, and the Postgraduate Innovation Project of Jiangsu University no. CX09B 13XZ and the Student’s Research Foundation of Jiangsu Universityno. 08A175.

References

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