Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2013, Article ID 961571, 5 pages
http://dx.doi.org/10.1155/2013/961571
Research Article
Some Limit Properties of the Harmonic Mean of Transition
Probabilities for Markov Chains in Markovian Environments
Indexed by Cayley’s Trees
Huilin Huang
College of Mathematics and Information Science, Wenzhou University, Zhejiang 325035, China
Correspondence should be addressed to Huilin Huang; huilin huang@sjtu.org
Received 7 August 2013; Revised 24 October 2013; Accepted 31 October 2013
Academic Editor: Onesimo Hernandez Lerma
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 some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a
homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to
study the tree-indexed processes in deterministic environments to the case of random enviroments.
1. Introduction
A tree 𝑇 is a graph which is connected and doesn’t contain
any circuits. Given any two vertices 𝛼 ≠ 𝛽 ∈ 𝑇, let 𝛼𝛽 be
the unique path connecting 𝛼 and 𝛽. Define the graph
distance 𝑑(𝛼, 𝛽) to be the number of edges contained in the
path 𝛼𝛽.
Let 𝑇 be an infinite tree with root 0. The set of all vertices
with distance 𝑛 from the root is called the 𝑛th generation
of 𝑇, which is denoted by 𝐿 𝑛 . We denote by 𝑇(𝑛) the union
of the first 𝑛 generations of 𝑇. For each vertex 𝑡, there is a
unique path from 0 to 𝑡 and |𝑡| for the number of edges on
this path. We denote the first predecessor of 𝑡 by 1𝑡. The
degree of a vertex is defined to be the number of neighbors
of it. If every vertex of the tree has degree 𝑑 + 1, we say it is
Cayley’s tree, which is denoted by 𝑇𝐶,𝑑 . Thus, the root vertex
has 𝑑 + 1 neighbors in the first generation and every other
vertex has 𝑑 neighbors in the next generation. For any two
vertices 𝑠 and 𝑡 of tree 𝑇, write 𝑠 ≤ 𝑡 if 𝑠 is on the unique
path from the root 0 to 𝑡. We denote by 𝑠 ∧ 𝑡 the farthest
vertex from 0 satisfying 𝑠 ∧ 𝑡 ≤ 𝑠 and 𝑠 ∧ 𝑡 ≤ 𝑡. We use the
notation 𝑋𝐴 = {𝑋𝑡 , 𝑡 ∈ 𝐴} and denote by |𝐴| the number of
vertices of 𝐴.
In the following, we always let 𝑇 denote the Cayley
tree 𝑇𝐶,𝑑 .
A tree-indexed Markov chain is the particular case of a
Markov random field on a tree. Kemeny et al. [1] and Spitzer
[2] introduced two special finite tree-indexed Markov chains
with finite transition matrix which is assumed to be positive
and reversible to its stationary distribution, and these treeindexed Markov chains ensure that the cylinder probabilities
are independent of the direction we travel along a path. In this
paper, we omit such assumption and adopt another version
of the definition of tree-indexed Markov chains which is put
forward by Benjamini and Peres [3]. Yang and Ye[4] extended
it to the case of nonhomogeneous Markov chains indexed by
infinite Cayley’s tree and we restate it here as follows.
Definition 1 (T-indexed nonhomogeneous Markov chains
(see [4])). Let 𝑇 be an infinite Cayley tree, X a finite state
space, and {𝑋𝑡 , 𝑡 ∈ 𝑇} a stochastic process defined on
probability space (Ω, 𝐹, P), which takes values in the finite
set X. Let
𝑝 = {𝑝 (𝑖) , 𝑖 ∈ X}
(1)
be a distribution on X and
𝑃𝑡 = (𝑃𝑡 (𝑗 | 𝑖)) ,
𝑖, 𝑗 ∈ X,
(2)
2
International Journal of Stochastic Analysis
a transition probability matrix on X2 . If, for any vertex 𝑡,
P (𝑋𝑡 = 𝑗 | 𝑋1𝑡 = 𝑖, 𝑋𝑠 = 𝑥𝑠 for 𝑡 ∧ 𝑠 ≤ 1𝑡)
= P (𝑋𝑡 = 𝑗 | 𝑋1𝑡 = 𝑖) = 𝑃𝑡 (𝑗 | 𝑖) ,
P (𝑋0 = 𝑖) = 𝑝 (𝑖)
∀𝑖, 𝑗 ∈ X,
∀𝑖 ∈ X,
(3)
(4)
then {𝑋𝑡 , 𝑡 ∈ 𝑇} will be called X-valued nonhomogeneous
Markov chains indexed by infinite Cayley’s tree with initial
distribution (1) and transition probability matrices {𝑃𝑡 , 𝑡 ∈
𝑇}.
The subject of tree-indexed processes has been deeply
studied and made abundant achievements. Benjamini and
Peres [3] have given the notion of the tree-indexed Markov
chains and studied the recurrence and ray-recurrence for
them. Berger and Ye [5] have studied the existence of entropy
rate for some stationary random fields on a homogeneous
tree. Ye and Berger [6, 7], by using Pemantle’s result [8]
and a combinatorial approach, have studied the ShannonMcMillan theorem with convergence in probability for a
PPG-invariant and ergodic random field on a homogeneous tree. Yang and Liu [9] and Yang [10] have studied
a strong law of large numbers for Markov chains fields
on a homogeneous tree (a particular case of tree-indexed
Markov chains and PPG-invariant random fields). Yang and
Ye [4] have established the Shannon-McMillan theorem for
nonhomogeneous Markov chains on a homogeneous tree.
Huang and Yang [11] has studied the strong law of large
numbers for finite homogeneous Markov chains indexed by
a uniformly bounded infinite tree.
The previous results are all about tree-indexed processes
in deterministic environments. Recently, we are interested
in random fields indexed by trees in random environments.
In the rest of this paper we formulate a model of Markov
chain indexed by trees in random environment especially
in Markovian environment and study some limit properties
of the harmonic mean of random transition probability for
finite Markov chains indexed by a homogeneous tree in
a nonhomogeneous Markovian environment. We are also
interested in the strong law of large numbers of Markov
chains indexed by trees in random environments which we
will prepare for in another paper.
Definition 2. Let 𝑇 be an infinite Cayley tree, X and Θ two
finite state spaces. Suppose that {𝜉𝑡 , 𝑡 ∈ 𝑇} is a Θ-valued
random field indexed by 𝑇, if, for any vertex 𝑡 ∈ 𝑇,
P (𝑋𝑡 = 𝑗 | 𝑋1𝑡 = 𝑖, 𝑋𝑠 = 𝑥𝑠 , for 𝑡 ∧ 𝑠 ≤ 1𝑡; 𝜉𝑟 , 𝑟 ∈ 𝑇)
= 𝑃𝜉1𝑡 (𝑖, 𝑗) ,
a.s.,
P (𝑋0 = 𝑖 | 𝜉𝑟 , 𝑟 ∈ 𝑇) = 𝑃 (𝑖 | 𝜉0 )
∀𝑖 ∈ X
(5)
Θ} and one-step transition probability matrices 𝐾𝑡 =
{𝐾𝑡 (𝛼, 𝛽), 𝛼, 𝛽 ∈ Θ, 𝑡 ∈ 𝑇}, we call {𝑋𝑡 , 𝑡 ∈ 𝑇} a Markov
chain indexed by tree 𝑇 in a nonhomogeneous Markovian
environment.
Remark 3. If every vertex has degree two, then our model of
Markov chain indexed by homogeneous tree with Markovian
environments is reduced to the model of Markov chain in
Markovian environments which was introduced by Cogburn
[12] in spirit.
Remark 4. We also point out that our model is different
from the tree-valued random walk in random environment
(RWRE) that is studied by Pemantle and Peres [13] and
Hu and Shi [14]. For the model of RWRE on the trees
that they studied, the random environment process 𝜔 :=
(𝜔(𝑥, ⋅), 𝑥, 𝑦 ∈ 𝑇) is a family of i.i.d nondegenerate random vectors and the process 𝑋 = {𝑋𝑛 , 𝑛 ∈ Z+ } is a
nearest-neighbor walk satisfying some conditions. But in our
model,our environmental process 𝜉 = {𝜉𝑡 , 𝑡 ∈ 𝑇} can be any
Markov chain indexed by trees. Given 𝜉, the process {𝑋𝑡 , 𝑡 ∈
𝑇} is another Markov chain indexed by trees with the law 𝑃𝜉 .
In this paper we assume that {𝜉𝑡 , 𝑡 ∈ 𝑇} is a nonhomogeneous 𝑇-indexed Markov chain on state space Θ.
The probability of going from 𝑖 to 𝑗 in one step in
the 𝜃th environment is denoted by 𝑃𝜃 (𝑖, 𝑗). We also
suppose that the one-step transition probability of going
from 𝛼 to 𝛽 for nonhomogeneous 𝑇-indexed Markov
chain {𝜉𝑡 , 𝑡 ∈ 𝑇} is 𝐾𝑡 (𝛼, 𝛽). In this case, {𝜉𝑡 , 𝑋𝑡 , 𝑡 ∈ 𝑇} is
a Markov chain indexed by 𝑇 with initial distribution 𝑞 =
(𝑞(𝜃, 𝑖)) and one-step transition on Θ × X determined by
𝑃𝑡 (𝛼, 𝑖; 𝛽, 𝑗) = 𝐾𝑡 (𝛼, 𝛽) 𝑃𝛼 (𝑖, 𝑗) ,
where 𝑞(𝜃, 𝑖) = P(𝜉0 = 𝜃, 𝑋0 = 𝑖). Then {𝜉𝑡 , 𝑋𝑡 , 𝑡 ∈ 𝑇} will be
called the bichain indexed by tree 𝑇. Obviously, we have
(𝑛)
(𝑛)
(𝑛)
(𝑛)
P (𝜉𝑇 = 𝛼𝑇 , 𝑋𝑇 = 𝑥𝑇 )
= 𝑞 (𝛼0 , 𝑥0 )
∏ 𝑃𝑡 (𝛼1𝑡 , 𝑥1𝑡 ; 𝛼𝑡 , 𝑥𝑡 ) .
(8)
𝑡∈𝑇(𝑛) \{0}
2. Main Results
For every finite 𝑛 ∈ N, let {𝑋𝑡 , 𝑡 ∈ 𝑇} be a Markov
chain indexed by an infinite Cayley tree 𝑇 in Markovian
environment {𝜉𝑡 , 𝑡 ∈ 𝑇}, which is defined as in Definition 2.
Now we suppose that 𝑔𝑡 (𝛼, 𝑖, 𝛽, 𝑗) are functions defined
on Θ × X × Θ × X. Let 𝜆 be a real number, 𝐿 0 = {0}, F𝑛 =
(𝑛)
(𝑛)
𝜎(𝜉𝑇 , 𝑋𝑇 ); now we define a stochastic sequence as follows:
(6)
for each 𝑖, 𝑗 ∈ X, where 𝑃𝑥 = (𝑃𝑥 (𝑖, 𝑗))𝑖,𝑗∈X , 𝑥 ∈ Θ is a family
of stochastic matrices. Then we call {𝑋𝑡 , 𝑡 ∈ 𝑇} a Markov
chain indexed by tree 𝑇 in a random environment {𝜉𝑡 , 𝑡 ∈
𝑇}. The 𝜉𝑡󸀠 𝑠 are called the environmental process or control
process indexed by tree 𝑇. Moreover, if {𝜉𝑡 , 𝑡 ∈ 𝑇} is a Tindexed Markov chain with initial distribution 𝜇 = {𝜇(𝜃), 𝜃 ∈
(7)
𝜑𝑛 (𝜆, 𝜔) =
𝑒𝜆 ∑𝑡∈𝑇(𝑛) \{0} 𝑔𝑡 (𝜉1 𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 )
.
∏𝑡∈𝑇(𝑛) \{0} 𝐸 [𝑒𝜆𝑔𝑡 (𝜉1 𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]
(9)
At first we come to prove the following fact.
Lemma 5. {𝜑𝑛 (𝜆, 𝜔), F𝑛 , 𝑛 ≥ 1} is a nonnegative martingale.
International Journal of Stochastic Analysis
3
if there exist two positive constants 𝑐 and 𝑚 such that
Proof of Lemma 5. Obviously, we have
(𝑛−1)
P (𝜉𝐿 𝑛 = 𝛼𝐿 𝑛 , 𝑋𝐿 𝑛 = 𝑥𝐿 𝑛 | 𝜉𝑇
=𝛼
𝑇(𝑛−1)
,𝑋
𝑇(𝑛−1)
=𝑥
(𝑛)
=
𝑇(𝑛−1)
(𝑛)
1
lim sup 󵄨 (𝑛) 󵄨 ∑ 𝑒𝑐/𝑎𝑡 = 𝑚 < ∞.
󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
𝑡∈𝑇(𝑛) \{0}
)
(𝑛)
(𝑛)
P (𝜉𝑇 = 𝛼𝑇 , 𝑋𝑇 = 𝑥𝑇 )
(𝑛−1)
P (𝜉𝑇
(𝑛−1)
= 𝛼𝑇
(𝑛−1)
, 𝑋𝑇
(10)
(𝑛−1)
= 𝑥𝑇
𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 ) = 𝑃𝑡 (𝜉𝑡 , 𝑋𝑡 | 𝜉1𝑡 , 𝑋1𝑡 ) ;
Here, the second equation holds because of the fact
that {𝜉𝑡 , 𝑋𝑡 , 𝑡 ∈ 𝑇} is a bichain indexed by tree 𝑇 and (8) is
being used. Furthermore, we have
| F𝑛−1 ]
󵄨󵄨 (𝑛) 󵄨󵄨
󵄨󵄨𝑇 󵄨󵄨
1
󵄨
󵄨
=
lim
−1
𝑛→∞
𝑀𝑁
∑𝑡∈𝑇(𝑛) \{0} 𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 )
𝜑𝑛 (𝜆, 𝜔) = 𝜑 (𝜆, 𝜔) < ∞
lim
𝑛
𝛼𝐿 𝑛 ,𝑥𝐿 𝑛
(𝑛−1)
(𝑛−1)
, 𝑋𝑇
)
lim
𝑛→∞
(11)
× P (𝜉𝑡 = 𝛼𝑡 , 𝑋𝑡 = 𝑥𝑡 | 𝜉1𝑡 , 𝑋1𝑡 )
= ∏
∑
𝑒
lim sup
𝑛→∞
× P (𝜉𝑡 = 𝛼𝑡 , 𝑋𝑡 = 𝑥𝑡 | 𝜉1𝑡 , 𝑋1𝑡 )
(19)
ln 𝜑𝑛 (𝜆, 𝜔)
≤ 0 a.s.
󵄨󵄨 (𝑛) 󵄨󵄨
󵄨󵄨𝑇 󵄨󵄨
(20)
We arrive at
a.s.
1
lim sup 󵄨 (𝑛) 󵄨
𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨
𝑡∈𝐿 𝑛
On the other hand, we also have
×
𝜑𝑛 (𝜆, 𝜔) = 𝜑𝑛−1 (𝜆, 𝜔)
∑
𝑡∈𝑇(𝑛) \{0}
𝑒𝜆 ∑𝑡∈𝐿 𝑛 𝑔𝑡 (𝜉1 𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 )
.
∏𝑡∈𝐿 𝑛 𝐸 [𝑒𝜆𝑔𝑡 (𝜉1𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]
(12)
Combining (11) and (12), we get
𝐸 [𝜑𝑛 (𝜆, 𝜔) | F𝑛−1 ] = 𝜑𝑛−1 (𝜆, 𝜔)
a.s.
(13)
Thus, we complete the proof of Lemma 5.
Theorem 6. Let {𝑋𝑡 , 𝑡 ∈ 𝑇} be a Markov chain indexed by
an infinite Cayley tree 𝑇 in a nonhomogeneous Markovian
environment {𝜉𝑡 , 𝑡 ∈ 𝑇}. Suppose that the initial distribution
and the transition probability functions satisfy
𝑞 (𝛼0 , 𝑥0 ) > 0,
(18)
which implies that
𝑡∈𝐿 𝑛 (𝛼𝑡 ,𝑥𝑡 )∈Θ×X
×
a.s.,
ln 𝜑𝑛 (𝜆, 𝜔)
= 0 a.s.,
󵄨󵄨 (𝑛) 󵄨󵄨
󵄨󵄨𝑇 󵄨󵄨
𝜆𝑔𝑡 (𝜉1 𝑡 ,𝑋1𝑡 ,𝛼𝑡 ,𝑥𝑡 )
= ∏ 𝐸 [𝑒𝜆𝑔𝑡 (𝜉1 𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]
(17)
so that
= ∑ ∏ 𝑒𝜆𝑔𝑡 (𝜉1𝑡 ,𝑋1 𝑡 ,𝛼𝑡 ,𝑥𝑡 )
𝛼𝐿 𝑛 ,𝑥𝐿 𝑛 𝑡∈𝐿 𝑛
a.s.
Proof. By Lemma 5, we have known that {𝜑𝑛 (𝜆, 𝜔), F𝑛 , 𝑛 ≥
1} is a nonnegative martingale. According to Doob martingale convergence theorem, we have
= ∑ 𝑒𝜆 ∑𝑡∈𝐿 𝑛 𝑔𝑡 (𝜉1 𝑡 ,𝑋1𝑡 ,𝛼𝑡 ,𝑥𝑡 )
× P (𝜉𝐿 𝑛 = 𝛼𝐿 𝑛 , 𝑋𝐿 𝑛 = 𝑥𝐿 𝑛 | 𝜉𝑇
(16)
then we have
𝑡∈𝐿 𝑛
𝐸 [𝑒
Denote |Θ| = 𝑀, |X| = 𝑁, and
)
= ∏ P (𝜉𝑡 = 𝛼𝑡 , 𝑋𝑡 = 𝑥𝑡 | 𝜉1𝑡 = 𝛼1𝑡 , 𝑋1𝑡 = 𝑥1𝑡 ) .
𝜆 ∑𝑡∈𝐿 𝑛 𝑔𝑡 (𝜉1 𝑡 ,𝑋1𝑡 ,𝜉𝑡 ,𝑋𝑡 )
(15)
𝑃𝑡 (𝛼, 𝑖; 𝜃, 𝑗) > 0,
for ∀𝛼0 , 𝛼, 𝜃 ∈ Θ; 𝑥0 , 𝑖, 𝑗 ∈ X,
𝑎𝑡 = min {𝑃𝑡 (𝛼, 𝑖; 𝜃, 𝑗) , 𝛼, 𝜃 ∈ Θ; 𝑖, 𝑗 ∈ X} , 𝑡 ∈ 𝑇 \ {0} ,
(14)
{𝜆𝑔𝑡 (𝜉1𝑡 , 𝑋1𝑡 , 𝜉𝑡 , 𝑋𝑡 )
− ln [𝐸 [𝑒𝜆𝑔𝑡 (𝜉1𝑡 ,𝑋1 𝑡 ,𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]]} ≤ 0
a.s.
(21)
Combining (20) with the inequalities ln 𝑥 ≤ 𝑥 − 1
(𝑥 > 0) and 0 ≤ 𝑒𝑥 − 1 − 𝑥 ≤ 2−1 𝑥2 𝑒|𝑥| and
taking 𝑔𝑡 (𝜉1𝑡 , 𝑋1𝑡 , 𝜉𝑡 , 𝑋𝑡 ) = 𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 )−1 , it follows
that
1
−1
lim sup 󵄨 (𝑛) 󵄨 ∑ [𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 ) 𝜆 − 𝜆𝑀𝑁]
󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
(𝑛)
𝑡∈𝑇 \{0}
1
≤ lim sup 󵄨 (𝑛) 󵄨
󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
×
∑
𝑡∈𝑇(𝑛) \{0}
−1
{ln [𝐸 [𝑒𝜆𝑃𝑡 (𝜉1𝑡 ,𝑋1𝑡 ;𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]]
− 𝜆𝑀𝑁}
4
International Journal of Stochastic Analysis
1
≤ lim sup 󵄨 (𝑛) 󵄨
𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨
×
If −𝑐 < 𝜆 < 0, similar to the analysis of inequality (24), by
using (15), (22), and (23) again, we can arrive at
1
−1
lim inf 󵄨 (𝑛) 󵄨 ∑ [𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 ) − 𝑀𝑁]
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨
󵄨
󵄨 𝑡∈𝑇(𝑛) \{0}
−1
∑
𝑡∈𝑇(𝑛) \{0}
{𝐸 [𝑒𝜆𝑃𝑡 (𝜉1𝑡 ,𝑋1𝑡 ;𝜉𝑡 ,𝑋𝑡 ) | 𝜉1𝑡 , 𝑋1𝑡 ]
− 1 − 𝜆𝑀𝑁}
≥
1
≤ lim sup 󵄨 (𝑛) 󵄨
𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨
×
∑
∑ ∑ 𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝛼, 𝑖)
× [𝑒
≤
1
𝜆
lim sup 󵄨 (𝑛) 󵄨
󵄨
2 𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
1
1 𝑒
𝜆𝑀𝑁
(
)
lim sup
∑
2 𝑛 → ∞ 󵄨󵄨󵄨󵄨𝑇(𝑛) 󵄨󵄨󵄨󵄨 𝑡∈𝑇(𝑛) \{0} 𝑎𝑡 𝑒𝑐
≥
𝜆𝑀𝑁
𝑚 a.s.
2𝑒 (𝜆 + 𝑐)
∑∑
1
1 |𝜆|/𝑎𝑡
𝜆2 𝑀𝑁
𝑒
lim sup 󵄨 (𝑛) 󵄨 ∑
2
𝑎
𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨
𝑡∈𝑇(𝑛) \{0} 𝑡
×
𝑥>0
0 < 𝛾 < 1.
(23)
𝜆𝑀𝑁
1
1 𝜆/𝑎𝑡
𝑒
lim sup
∑
2 𝑛 → ∞ 󵄨󵄨󵄨󵄨𝑇(𝑛) 󵄨󵄨󵄨󵄨 𝑡∈𝑇(𝑛) \{0} 𝑎𝑡
1
1 𝜆−𝑐 1/𝑎𝑡 𝑐/𝑎𝑡
𝜆𝑀𝑁
=
(𝑒 ) 𝑒
lim sup 󵄨 (𝑛) 󵄨 ∑
2 𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨 𝑡∈𝑇(𝑛) \{0} 𝑎𝑡
𝑡∈𝑇(𝑛) \{0}
(24)
a.s.
−1
[𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 )
𝑝 (𝑥0 ) > 0,
𝑃𝑡 (𝑖, 𝑗) > 0,
𝑡 ∈ 𝑇 \ {0} ,
1
lim sup 󵄨 (𝑛) 󵄨 ∑ 𝑒𝑐/𝑎𝑡 = 𝑚 < ∞.
𝑛 → ∞ 󵄨󵄨󵄨𝑇 󵄨󵄨󵄨
𝑡∈𝑇(𝑛) \{0}
1
lim sup 󵄨 (𝑛) 󵄨
󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
∑
− 𝑀𝑁] ≥ 0 a.s.
(28)
if there exist two positive constants 𝑐 and 𝑚 such that
Here, the second equation holds because by letting 𝜆 →
0+ in inequality (24), we get
×
−1
Combining (25) and (27), we obtain that our assertion (17) is
true.
𝑎𝑡 = min {𝑃𝑡 (𝑖, 𝑗) , 𝑖, 𝑗 ∈ X} ,
1
−1
lim sup 󵄨 (𝑛) 󵄨 ∑ [𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 ) − 𝑀𝑁]
󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨󵄨
(𝑛)
𝑡∈𝑇 \{0}
𝜆𝑀𝑁
𝑚
≤
2𝑒 (𝑐 − 𝜆)
[𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝜉𝑡 , 𝑋𝑡 )
Corollary 7. Let {𝑋𝑡 , 𝑡 ∈ 𝑇} be a nonhomogeneous Markov
chain indexed by an infinite Cayley tree 𝑇. Suppose that the
initial distribution and the transition probability functions
satisfy
Let 0 < 𝜆 < 𝑐. It follows from (15), (22), and (23) that
≤
𝑒𝑐/𝑎𝑡
(27)
a.s.
Note that the following elementary fact holds:
𝑒−1
,
ln 𝛾
∑
𝑡∈𝑇(𝑛) \{0}
(22)
max {𝑥𝛾𝑥 } = −
(26)
1
lim inf 󵄨 (𝑛) 󵄨
𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨
󵄨
󵄨
1
𝑒|𝜆|/𝑃𝑡 (𝜉1𝑡 ,𝑋1𝑡 ;𝛼,𝑖)
𝑃
(𝜉
,
𝑋
;
𝛼,
𝑖)
(𝑛)
𝑡
1𝑡
1𝑡
𝑡∈𝑇 \{0} 𝛼∈Θ 𝑖∈X
∑
1/𝑎𝑡
Letting 𝜆 → 0− in inequality (26), we get
2
×
≤
𝜆
−1−
]
𝑃𝑡 (𝜉1𝑡 , 𝑋1𝑡 ; 𝛼, 𝑖)
−𝜆
≥
𝑡∈𝑇(𝑛) \{0} 𝛼∈Θ 𝑖∈X
𝜆/𝑃𝑡 (𝜉1𝑡 ,𝑋1𝑡 ;𝛼,𝑖)
1
1 −𝜆/𝑎𝑡
𝜆𝑀𝑁
𝑒
lim sup 󵄨 (𝑛) 󵄨 ∑
󵄨
󵄨
2 𝑛 → ∞ 󵄨󵄨𝑇 󵄨󵄨 𝑡∈𝑇(𝑛) \{0} 𝑎𝑡
− 𝑀𝑁] ≤ 0 a.s.
(25)
Denote |X| = 𝑁; then we have
󵄨󵄨 (𝑛) 󵄨󵄨
󵄨󵄨𝑇 󵄨󵄨
1
󵄨
󵄨
=
lim
−1
𝑛→∞
𝑁
∑𝑡∈𝑇(𝑛) \{0} 𝑃𝑡 (𝑋𝑡 | 𝑋1𝑡 )
(29)
a.s.
(30)
Proof. If we take Θ = {𝜃}, that is, |Θ| = 1, then the model of
Markov chain indexed by tree 𝑇 in Markovian environment
reduces to the formulation of a nonhomogeneous Markov
chain indexed by tree 𝑇. Then we arrive at our conclusion
(30) directly from Theorem 6.
Corollary 8 (see [15]). Let {𝑋𝑛 , 𝑛 ≥ 0} be a Markov chain in
a nonhomogeneous Markovian environment {𝜉𝑛 , 𝑛 ≥ 0}. Suppose that the initial distribution and the transition probability
functions satisfy
𝑞 (𝛼0 , 𝑥0 ) > 0,
𝑃𝑛 (𝛼, 𝑖; 𝜃, 𝑗) > 0,
𝑎𝑛 = min {𝑃𝑛 (𝛼, 𝑖; 𝜃, 𝑗) , 𝛼, 𝜃 ∈ Θ; 𝑖, 𝑗 ∈ X} ,
𝑛 ≥ 1,
(31)
International Journal of Stochastic Analysis
5
if there exist two positive constants 𝑐 and 𝑚 such that
1 𝑛
lim sup ∑ 𝑒𝑐/𝑎𝑘 = 𝑚 < ∞.
𝑛 → ∞ 𝑛 𝑘=1
(32)
Denote |Θ| = 𝑀, |X| = 𝑁, and
𝑃𝑘 (𝜉𝑘−1 , 𝑋𝑘−1 ; 𝜉𝑘 , 𝑋𝑘 ) = 𝑃𝑘 (𝜉𝑘 , 𝑋𝑘 | 𝜉𝑘−1 , 𝑋𝑘−1 ) ;
(33)
then we have
lim
𝑛→∞
𝑛
∑𝑛𝑘=1
−1
𝑃𝑘 (𝜉𝑘−1 , 𝑋𝑘−1 ; 𝜉𝑘 , 𝑋𝑘 )
=
1
𝑀𝑁
a.s.
(34)
Proof. If every vertex of the tree 𝑇 has degree 2,
then the nonhomogeneous Markov chain indexed by
tree 𝑇 degenerates into the nonhomogeneous Markov chain
on line; thus, this corollary can be obtained from Theorem 6
directly.
Acknowledgment
This work was supported by the National Natural Science
Foundation of China no. 11201344. The author declares that
there is no conflict of interests regarding the publication of
this paper.
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