A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system
We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorem...
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| author | Kangkang, Wang Канґканг, Ванг |
| author_facet | Kangkang, Wang Канґканг, Ванг |
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| description | We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed
by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale.
As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states,
ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree. |
| first_indexed | 2026-03-24T02:30:45Z |
| format | Article |
| fulltext |
UDC 519.21
Kangkang Wang (School Math. and Phys., Jiangsu Univ. Sci. and Technology, Zhenjiang, China)
A CLASS OF STRONG LIMIT THEOREMS
FOR NONHOMOGENEOUS MARKOV CHAINS
INDEXED BY A GENERALIZED BETHE TREE
ON A GENERALIZED RANDOM SELECTION SYSTEM*
ПРО ОДИН КЛАС СИЛЬНИХ ГРАНИЧНИХ ТЕОРЕМ
ДЛЯ НЕОДНОРIДНИХ МАРКОВСЬКИХ ЛАНЦЮЖКIВ,
ЩО ПРОIНДЕКСОВАНI УЗАГАЛЬНЕНИМ ДЕРЕВОМ БЕТЕ
НА УЗАГАЛЬНЕНIЙ СИСТЕМI ВИПАДКОВОГО ВИБОРУ
We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed
by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale.
As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states,
ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree.
Вивчаються сильнi граничнi теореми для послiдовностi функцiй двох змiнних неоднорiдного марков-
ського ланцюжка, що проiндексований узагальненим деревом Бете на узагальненiй системi випадкового
вибору, шляхом побудови невiд’ємного мартингала. Як наслiдок, узагальнено результати Янга та Є i
отримано деякi граничнi теореми для частот станiв, упорядкованих пар та умовного сподiвання функцiї
двох змiнних на деревi Келi.
1. Introduction and definition. Let T be a tree which is infinite, connected and contains
no circuits. Given any two vertices x 6= y ∈ T, there exists an unique path x =
x1, x2, . . . , xm = y from x to y with x1, x2, . . . , xm distinct. The distance between x
and y is defined to m−1, the number of edges in the path connecting x and y. To index
the vertices on T, we first assign a vertex as the ,,root”and label it as O. A vertex is said
to be on the n th level if the path linking it to the root has n edges. The root O is also
said to be on the 0th level.
Definition 1. Let T be a tree with root O, and let {Nn, n ≥ 1} be a sequence of
positive integers. T is said to be a generalized Bethe tree or a generalized Cayley tree if
each vertex on the nth level has Nn+1 branches to the n+1th level. For example, when
N1 = N + 1 ≥ 2 and Nn = N, n ≥ 2, T is rooted Bethe tree (a homogeneous tree)
TB,N on which each vertex has N + 1 neighboring vertices (TB,2 drawn in Figure),
and when Nn = N ≥ 1, n ≥ 1, T is rooted Cayley tree TC,N on which each vertex has
N branches to the next level.
In the following, we always assume that T is a generalized Bethe tree and denote by
T (n) the subgraph of T containing the vertices from level 0 (the root) to level n. We use
(n, j), 1 ≤ j ≤ N1 . . . Nn, n ≥ 1, to denote the jth vertex at the nth level and denote
by |B| the number of vertices in the subgraph B, Lnm the set of all vertices from level
m to level n, Ln the set of all vertices on level n. It is easy to see that for n ≥ 1,
|T (n)| =
n∑
m=0
N0 . . . Nm = 1 +
n∑
m=1
N1 . . . Nm. (1)
Let S = {s0, s1, s2, . . .}, Ω = ST , ω = ω(·) ∈ Ω, where ω(·) is a function defined on
T and takes values in S, and F be the smallest Borel field containing all cylinder sets in
*The work is supported by the Project of Higher Schools’ Natural Science Basic Research of Jiangsu
Province of China (09KJD110002).
c© KANGKANG WANG, 2011
1336 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1337
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Bethe tree TB,2
Ω. Let X = {Xt, t ∈ T} be the coordinate stochastic process defined on the measurable
space (Ω,F) (see [1, p. 412]); that is, for any ω = {ω(t), t ∈ T}, define
Xt(ω) = ω(t), t ∈ T. (2)
Let µ be an arbitrary probability measure defined on (Ω,F). Denote
XT (n) ∆
=
{
Xt, t ∈ T (n)
}
, µ
(
XT (n)
= xT
(n))
= µ
(
xT
(n))
, (3)
where ω(t) is in fact the sample point function with respect to t, X = {Xt, t ∈ T}
is a stochastic process defined on the tree T, that is, X = {Xt, t ∈ T} is a se-
quence of random variables defined on all the vertices of T
(
i.e., {Xt, t ∈ T} =
= {X0,1, X1,1, X1,2, ... , X1,N0N1 , X2,1, ... , X2,N0N1N2 , ... , Xm,1, ... , Xm,N0...Nm , ...}
)
.
We denote by xT
(n)
the realization of the stochastic process XT (n)
. XT (n)
stands for
the sequence of the random variables defined on all the vertices from the root to level
n on the tree T . x0,1 is the realization of X0,1 which is the random variable defined on
the root.
Now we give a definition of Markov chains field on the tree T by using the cylinder
distribution directly, which is a natural extension of the classical definition of Markov
chains (see [2]).
Definition 2. Let {Pn = Pn(j|i), i, j ∈ S, n ≥ 1} be stochastic matrices on S2,
p = (p(s0), p(s1), p(s2), . . .) be a distribution on S, and µP be a measure on (Ω,F). If
µP (x0,1) = p(x0,1), (4)
µP (xT
(n)
) = p(x0,1)
n−1∏
m=0
N0...Nm∏
i=1
Nm+1i∏
j=Nm+1(i−1)+1
Pm+1(xm+1,j |xm,i), n ≥ 1. (5)
Then µP will be called a Markov chains field on the tree T determined by the stochastic
matrices Pn and the distribution p.
The tree model have recently drawn the increasing interest from specialists in
physics, probability and information theory. For the early studies on Markov chains
fields on trees see Spitzer [3]. Benjamini and Peres [4] have given the notion of the
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1338 KANGKANG WANG
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. Pemantle [6] proved a mixing property and a weak law of
large numbers for a PPG-invariant and ergodic random field on a homogeneous tree. Ye
and Berger [7], by using Pemantle’s result 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 [8, 9] and Liu [1] have studied
strong laws of large numbers for the frequency of occurrence of states for Markov chains
field on the Bethe tree and the generalized Bethe tree. Yang and Ye [2] have discussed
the strong limit theorems for nonhomogeneous Markov chain indexed by the homoge-
neous tree. Shi and Yang [10] have investigated a limit property of random transition
probability for a nonhomogeneous Markov chain indexed by a tree.
In this paper, we study a class of strong limit theorems for a bivariate function se-
quence for nonhomogeneous Markov chains field indexed by the generalized Bethe tree
on the the generalized random selection system by constructing a nonnegative martin-
gale. As corollaries, we generalize Yang and Ye’s results (see [2, 8]) and obtain some
limit theorems for frequencies of states, ordered couples of states, the harmonic mean
of the transition probabilities of the nonhomogeneous Markov chain and the conditional
expectation on Cayley tree.
Definition 3. Let {fm,i(x0,1, x1,1, . . . , x1,N0N1 , . . . , xm,1, . . . , xm,i−1), 0 ≤ m ≤
≤ n, 1 ≤ i ≤ N0 . . . Nm} be a series of real-valued functions defined on ST
(n)
,
n = 1, 2, . . . , which take values in an arbitrary interval [a, b] (a, b ∈ R). Denote
Y0 = f0,1 = 1,
Ym,i = fm,i(X0,1, X1,1, . . . , X1,N0N1
, . . . , Xm,1, . . . , Xm,i−1),
1 ≤ m ≤ n, 2 ≤ i ≤ N0 . . . Nm,
Ym+1,1 = fm+1,1(X0,1, X1,1, . . . , X1,N0N1 , . . . , Xm,1, . . . , Xm,N0...Nm),
1 ≤ m ≤ n− 1. (6)
We call {Ym,i, 0 ≤ m ≤ n, 1 ≤ i ≤ N0 . . . Nm} as the generalized random selection
system on the generalized Bethe trees (the traditional random selection system takes
values in the set {0, 1}).
We first explain the conception of the traditional random selection, which is the cru-
cial part of the gambling system. We give a set of real-valued functions fn(x1, . . . , xn)
defined on Sn, n = 1, 2, . . . , which will be called the random selection function if they
take values in a two-valued set {0, 1}. Then let
Y1 = y (y is an arbitrary real number),
Yn+1 = fn(X1, . . . , Xn), n ≥ 1,
where {Yn, n ≥ 1} be called as the gambling system (the random selection system). Let
δi(j) be the Kronecker delta function on S, that is for i, j ∈ S
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1339
δi(j) =
0, i 6= j,
1, i = j.
In order to explain the real meaning of the notion of the random selection, we con-
sider the traditional gambling model. Let {Xn, n ≥ 0} be a nonhomogeneous Markov
chain, and {gn(x, y), n ≥ 1} be a real-valued function sequence defined on S2. Inter-
pret Xn as the result of the n th trial, the type of which may change at each step. Let
µn = Yngn(Xn−1, Xn) denote the gain of the bettor at the n th trial, where Yn repre-
sents the bet size, gn(Xn−1, Xn) is determined by the gambling rules, and {Yn, n ≥ 0}
is called a gambling system or a random selection system. The bettor’s strategy is to
determine {Yn, n ≥ 1} by the results of the last trial. Let the entrance fee that the bettor
pays at the nth trial be bn. Also suppose that bn depends on Xn−1 as n ≥ 1, and b0 is
a constant. Thus
∑n
k=1
Ykgk(Xk−1, Xk) represents the total gain in the first n trials,∑n
k=1
bk the accumulated entrance fees, and
∑n
k=1
[
Ykgk(Xk−1, Xk)− bk
]
the accu-
mulated net gain. Motivated by the classical definition of ,,fairness”of game of chance
(see Kolmogorov [11]), we introduce the following definition:
Definition 4. The game is said to be fair, if for almost all ω ∈ {ω :
∑∞
k=1
Yk =
=∞}, the accumulated net gain in the first n trial is to be of smaller order of magnitude
than the accumulated stake
∑n
k=1
Yk as n tends to infinity, that is
lim
n→∞
1∑n
k=1
Yk
n∑
k=1
[Ykgk(Xk−1, Xk)− bk] = 0 a.s. on
{
ω :
∞∑
k=1
Yk =∞
}
.
We generalize the traditional gambling system to the case of the nonhomogeneous
Markov chain indexed by the generalized Bethe tree, and obtain the following conclu-
sion:
2. Main results.
Theorem 1. Let X = {Xt, t ∈ T} be a nonhomogeneous Markov chain indexed
by the generalized Bethe tree with the initial distribution and the transition matrices
defined as Definition 2. Let {gn(x, y), n ≥ 1} be a series of real-valued functions
defined on S2. Let {an, n ≥ 0} be a nonnegative stochastic sequence, denote α > 0,
Fn(ω) =
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
Ym,igm+1(Xm,i, Xm+1,j), (7)
Gn(ω) =
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Ym,igm+1(Xm,i, Xm+1,j)|Xm,i
]
, (8)
Hn(ω) =
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
exp{α |Ym,igm+1(Xm,i, Xm+1,j)|}|Xm,i
]
.
(9)
Put
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1340 KANGKANG WANG
D =
{
ω : lim
n→∞
an =∞, lim sup
n→∞
1
an
Hn(ω) <∞
}
, (10)
then
lim
n→∞
1
an
[Fn(ω)−Gn(ω)] = 0 µP -a.s., ω ∈ D. (11)
Proof. Consider the probability measure space (Ω,F , µP), letting λ be an arbitrary
constant, we construct
Tn(λ, ω) =
=
e
λ
[∑n−1
m=0
∑N0...Nm
i=1
∑Nm+1i
j=Nm+1(i−1)+1
Ym,igm+1(Xm,i,Xm+1,j)
]
∏n−1
m=0
∏N0...Nm
i=1
∏Nm+1i
j=Nm+1(i−1)+1
E[eλYm,igm+1(Xm,i,Xm+1,j)|Xm,i]
, (12)
n ≥ 1.
Noting that X = {Xt, t ∈ T} satisfies (5), we have
P(XLn = xLn |XT (n−1)
= xT
(n−1)
) =
µP (xT
(n)
)
µP (xT (n−1))
=
=
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
Pn(xn,j |xn−1,i). (13)
Denoting Fn = σ(XT (n)
), by (12), (13) and Markov’s property, we have
E [Tn(λ, ω)|Fn−1] =
= Tn−1(λ, ω)
E
[
e
λ
[∑N0...Nn−1
i=1
∑Nni
j=Nn(i−1)+1
Yn−1,ign(Xn−1,i,Xn,j)
]
|XT (n−1)
]
∏N0...Nn−1
i=1
∏Nni
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]
=
=
Tn−1(λ, ω)
∑
xLn∈SLn
e
λ
[
N0...Nn−1∑
i=1
Nni∑
j=Nn(i−1)+1
Yn−1,ign(Xn−1,i,xn,j)
]
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]
×
×
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
Pn(xn,j |Xn−1,i) =
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1341
= Tn−1(λ, ω)
∑
xLn∈SLn
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
eλYn−1,ign(Xn−1,i,xn,j)Pn(xn,j |Xn−1,i)
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]
=
= Tn−1(λ, ω)
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
∑
xn,j∈S
eλYn−1,ign(Xn−1,i,xn,j)Pn(xn,j |Xn−1,i)
N0...Nn−1∏
i=1
Nni∏
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]
=
= Tn−1(λ, ω)
∏N0...Nn−1
i=1
∏Nni
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]∏N0...Nn−1
i=1
∏Nni
j=Nn(i−1)+1
E[eλYn−1,ign(Xn−1,i,Xn,j)|Xn−1,i]
=
= Tn−1(λ, ω). (14)
Therefore, {Tn(λ, ω),Fn, n ≥ 1} is a nonnegative martingale. By Doob’s martingale
convergence theorem, we obtain
lim
n→∞
Tn(λ, ω) = T∞(λ, ω) <∞ µP -a.s. (15)
By the first equation lim
n→∞
an =∞ of (10) and (15) we have
lim sup
n→∞
1
an
lnTn(λ, ω) ≤ 0 µP -a.s., ω ∈ D. (16)
By (7), (12) and (16), we obtain
lim sup
n→∞
1
an
{
λFn(ω)−
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
ln E[eλYm,igm+1(Xm,i,Xm+1,j)|Xm,i]
}
≤
≤ 0 µP -a.s., ω ∈ D. (17)
By (8), (17) and the inequalities lnx ≤ x−1(x > 0), ex−1−x ≤ (1/2)x2e|x|, noticing
that
max
{
x2e−hx, x ≥ 0
}
=
4e−2
h2
, h > 0,
letting 0 < |λ| < α, we have
lim sup
n→∞
1
an
λ{Fn(ω)−Gn(ω)} ≤
≤ lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
{
ln E[eλYm,igm+1(Xm,i,Xm+1,j)|Xm,i]−
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1342 KANGKANG WANG
−E[λYm,igm+1(Xm,i, Xm+1,j)|Xm,i]
}
≤
≤ lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
{
E[eλYm,igm+1(Xm,i,Xm+1,j)|Xm,i]−
−1− E[λYm,igm+1(Xm,i, Xm+1,j)|Xm,i]
}
≤
≤ lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
λ2
2
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×e|λ||Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
=
=
λ2
2
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×e(|λ|−α)|Ym,igm+1(Xm,i,Xm+1,j)|eα|Ym,igm+1(Xm,i,Xm+1,j)|Xm,i
]
≤
≤ λ2
2
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
eα|Ym,igm+1(Xm,i,Xm+1,j)|4e−2
(|λ| − α)2
∣∣∣Xm,i
]
µP -a.s., ω ∈ D. (18)
Taking 0 < λ < α, dividing two sides of (18) by λ, we arrive at
lim sup
n→∞
1
an
{Fn(ω)−Gn(ω)} ≤ 2λe−2
(λ− α)2
×
× lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E[eα|Ym,igm+1(Xm,i,Xm+1,j)|Xm,i] <∞
µP -a.s., ω ∈ D. (19)
Since 2λe−2
/
(λ− α)2 → 0 as λ→ +0, by (19) we obtain
lim sup
n→∞
1
an
{Fn(ω)−Gn(ω)} ≤ 0 µP -a.s., ω ∈ D. (20)
Taking −α < λ < 0, dividing two sides of (18) by λ, we have
lim inf
n→∞
1
an
{
Fn(ω)−Gn(ω)
}
≥ 2λe−2
(λ+ α)2
lim sup
n→∞
1
an
Hn(ω) µP -a.s., ω ∈ D.
(21)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1343
Since 2λe−2
/
(λ+ α)2 → 0 as λ→ −0, by (21) we obtain
lim inf
n→∞
1
an
{Fn(ω)−Gn(ω)} ≥ 0 µP -a.s., ω ∈ D. (22)
Therefore, it follows from (20) and (22) that (11) holds.
Theorem 2. Let X = {Xt, t ∈ T} be a nonhomogeneous Markov chain indexed
by the generalized Bethe tree with the initial distribution and the transition matrices
defined as Definition 2. Let {gn(x, y), n ≥ 1}, {an, n ≥ 0}, Fn(ω) and Gn(ω) be all
defined as Theorem 1. Denote α > 0,
Bn(ω) =
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×eα|Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
. (23)
Put
L(ω) =
{
ω : lim
n→∞
an =∞, lim sup
n→∞
1
an
Bn(ω) <∞
}
, (24)
then
lim
n→∞
1
an
[
Fn(ω)−Gn(ω)
]
= 0 µP -a.s., ω ∈ L(ω). (25)
Proof. By the third inequality of (18) in the proof of Theorem 1, taking 0 < |λ| < α,
we arrive at
lim sup
n→∞
1
an
λ{Fn(ω)−Gn(ω)} ≤
≤ lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
λ2
2
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×e|λ||Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
≤
≤ λ2
2
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×eα|Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
<∞
µP -a.s., ω ∈ D. (26)
Take 0 < λ < α, dividing two sides of (26) by λ, we have
lim sup
n→∞
1
an
{Fn(ω)−Gn(ω)} ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1344 KANGKANG WANG
≤ λ
2
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×eα|Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
<∞ µP -a.s., ω ∈ D. (27)
Letting λ→ +0, we have by (27) that
lim sup
n→∞
1
an
{Fn(ω)−Gn(ω)} ≤ 0 µP -a.s., ω ∈ D. (28)
Taking −α < λ < 0 in (26), we similarly obtain
lim inf
n→∞
1
an
{Fn(ω)−Gn(ω)} ≥
≥ λ
2
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×eα|Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
µP -a.s., ω ∈ D.
Letting λ→ −0, we have
lim inf
n→∞
1
an
{Fn(ω)−Gn(ω)} ≥ 0 µP -a.s., ω ∈ D. (29)
It follows from (28) and (29) that (25) holds.
Corollary 1 [2]. Let X = {Xt, t ∈ T} be a nonhomogeneous Markov chain in-
dexed by a homogeneous tree TB,N . Let {gn(x, y), n ≥ 1}, {an, n ≥ 0} be defined as
Theorem 1. Denote α > 0,
Gn(ω) =
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
E
[
g2
m+1(Xm,i, Xm+1,j)×
×eα|gm+1(Xm,i,Xm+1,j)||Xm,i
]
. (30)
Put
J(ω) =
{
ω : lim
n→∞
an =∞, lim sup
n→∞
1
an
Gn(ω) <∞
}
, (31)
then
lim
n
1
an
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
{
gm+1(Xm,i, Xm+1,j)−
−E[gm+1(Xm,i, Xm+1,j)|Xm,i]
}
= 0 µP -a.s., ω ∈ J(ω). (32)
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A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1345
Proof. Letting N0 = 1, N1 = N + 1, Nn = N(n ≥ 2), Ym,i ≡ 1 in Theorem 2,
(30), (31) and (32) follow from (23), (24) and (25).
Remark. The corollary is Theorem 1 of Yang and Ye (see [2]). Letting Ym,i = 1 in
Theorem 1, it can be seen that the condition (9), (10) weakens the condition (30), (31) of
Theorem 1 in the paper of Yang and Ye. Correspondingly the conclusion is strengthened.
Corollary 2 [8]. Let X = {Xt, t ∈ T} be a nonhomogeneous Markov chain in-
dexed by the homogeneous tree. Let g(x, y) be a function defined on S2 taking values
in {0, 1}, {an, n ≥ 0} be defined as Theorem 1. Put
G(ω) =
{
ω : lim
n→∞
an =∞,
lim sup
n→∞
1
an
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
E
[
g(Xm,i, Xm+1,j)|Xm,i
]
<∞
}
, (33)
then
lim
n
1
an
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
{
g(Xm,i, Xm+1,j)−
−E[g(Xm,i, Xm+1,j)|Xm,i]
}
= 0 µP -a.s., ω ∈ G(ω). (34)
Proof. Letting gn(x, y) = g(x, y), n ≥ 1, Ym,i ≡ 1 in Theorem 2, by (23), (33)
and the definition of g(x, y), we have
lim sup
n→∞
1
an
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
E
[
Y 2
m,ig
2
m+1(Xm,i, Xm+1,j)×
×eα|Ym,igm+1(Xm,i,Xm+1,j)||Xm,i
]
≤
≤ lim sup
n→∞
1
an
eα
n−1∑
m=1
(N+1)Nm−1∑
i=1
Ni∑
j=N(i−1)+1
E
[
g(Xm,i, Xm+1,j)|Xm,i
]
<∞. (35)
Hence G(ω) ⊂ J(ω), (34) follows from Theorem 2.
Corollary 3. Let S = {1, 2, . . . , N}, and
βn = min{Pn(y|x), x, y ∈ S}, n ≥ 1. (36)
If there exists α > 0, such that
lim sup
n→∞
1
|T (n)|
n−1∑
m=0
e
α
βm+1
m+1∏
j=0
Nj = M <∞, (37)
then the harmonic mean of the transition probabilities {Pm+1(Xm+1,j |Xm,i), 0 ≤ m ≤
≤ n−1, 1 ≤ i ≤ N0 . . . Nm, Nm+1(i−1)+1 ≤ j ≤ Nm+1i} for the nonhomogeneous
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1346 KANGKANG WANG
Markov chain indexed by the generalized bethe tree converges to N−1 a.s., that is
lim
n→∞
|T (n)|∑n−1
m=0
∑N0...Nm
i=1
∑Nm+1i
j=Nm+1(i−1)+1
Pm+1(Xm+1,j |Xm,i)
−1
=
1
N
µP -a.s.
(38)
Proof. Letting an(ω) = |T (n)|, gm+1(Xm,i, Xm+1,j) = Pm+1(Xm+1,j |Xm,i)
−1,
Ym,i ≡ 1 in Theorem 1, by (9), (10), (36) and (37) we have
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
exp{α |Ym,igm+1(Xm,i, Xm+1,j)|}|Xm,i
]
=
= lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E[exp{α
∣∣Pm+1(Xm+1,j |Xm,i)
−1
∣∣}|Xm,i]≤
≤ lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E[e
α
βm+1 |Xm,i] =
= lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0 . . . NmNm+1e
α
βm+1 =
= lim sup
n→∞
1
|T (n)|
n−1∑
m=0
e
α
βm+1
m+1∏
j=0
Nj = M <∞. (39)
By (10) and (39) we obtain D = Ω. Noticing that
E
[
gm+1(Xm,i, Xm+1,j)|Xm,i
]
= E
[
Pm+1(Xm+1,j |Xm,i)
−1|Xm,i
]
=
=
∑
xm+1,j∈S
Pm+1(xm+1,j |Xm,i)
−1Pm+1(xm+1,j |Xm,i) = N. (40)
By (11) and (40), we arrive at
lim
n→∞
1
an
[Fn(ω)−Gn(ω)] =
= lim
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
[Pm+1(Xm+1,j |Xm,i)
−1 −N ] =
= lim
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
Pm+1(Xm+1,j |Xm,i)
−1−
− lim
n→∞
|T (n)| − 1
|T (n)|
N =
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A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1347
= lim
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
Pm+1(Xm+1,j |Xm,i)
−1 −N = 0.
Hence, (38) follows from the above equation.
Remark. The corollary is a generalization of Theorem 1 of Shi and Yang (see [10]).
3. Derivation results. In the Definition 2, if for all n,
Pn = P = (P (y|x)) ∀x, y ∈ S. (41)
X = {Xσ, σ ∈ T} will be also called S-valued homogeneous Markov chain indexed by
a generalized Bethe tree. At the moment, we have
µP (x0,1) = p(x0,1), (42)
µP (xT
(n)
) = p(x0,1)
n−1∏
m=0
N0...Nm∏
i=1
Nm+1i∏
j=Nm+1(i−1)+1
P (xm+1,j |xm,i), n ≥ 1. (43)
Theorem 3. Let X = {Xt, t ∈ T} be a homogeneous Markov chain indexed by
the generalized Bethe tree, g(x, y), Fn(ω) and Gn(ω) be defined as before.
{
Ym,i, 0 ≤
≤ m ≤ n, 1 ≤ i ≤ N0 . . . Nm
}
take values in a real-valued interval [a, b], where
a, b ∈ R. Denote M = max{|a|, |b|}, if∑
l∈S
∑
k∈S
exp{αM |g(k, l)|}P (l|k) <∞. (44)
Then
lim
n→∞
1
|T (n)|
[Fn(ω)−Gn(ω)] = 0 µP -a.s. (45)
Proof. Letting an = |T (n)| in Theorem 1, by (10) we have
lim sup
n→∞
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
E
[
exp{α |Ym,ig(Xm,i, Xm+1,j)|}|Xm,i
]
=
= lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
∑
xm+1,j∈S
exp{α |Ym,ig(Xm,i, xm+1,j)|}×
×P (xm+1,j |Xm,i) =
= lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
∑
l∈S
∑
k∈S
δk(Xm,i) exp{α |Ym,ig(k, l)|}×
×P (l|k) ≤ lim sup
n→∞
1
|T (n)|
n−1∑
m=0
N0...Nm∑
i=1
Nm+1i∑
j=Nm+1(i−1)+1
∑
l∈S
∑
k∈S
exp{αM |g(k, l)|}×
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1348 KANGKANG WANG
×P (l|k) ≤
∑
l∈S
∑
k∈S
lim sup
n→∞
|T (n)| − 1
|T (n)|
exp{αM |g(k, l)|}P (l|k) =
=
∑
l∈S
∑
k∈S
exp{αM |g(k, l)|}P (l|k). (46)
By (44) and (46), we have D = Ω. Therefore, (45) follows from Theorem 1.
Corollary 4 [2]. Let X = {Xt, t ∈ T} be a homogeneous Markov chain indexed
by a Cayley tree TC.N , Sn(k, l) be the number of couple (k, j) in the set of random
couple {(Xm,i, Xm+1,j), 0 ≤ m ≤ n − 1, 1 ≤ i ≤ Nm, N(i − 1) + 1 ≤ j ≤ Ni},
Sn(k) be the number of k in the set of random variables X = {Xt, t ∈ T (n)}. Then
lim
n→∞
[
Sn(k, l)
|T (n)|
− Sn−1(k)
|T (n−1)|
P (l|k)
]
= 0 µP -a.s. (47)
Proof. Letting an = |T (n)|, gn(x, y) = g(x, y) = Ik(x)Ij(y), n ≥ 1, N0 = 1,
Nn = N, n ≥ 1, Ym,i ≡ 1 in Theorem 1, we have by (10) that
lim sup
n
1
an
n−1∑
m=0
N0...Nm∑
i=1
Nm+!i∑
j=Nm+!(i−1)+1
E
[
exp{α |Ym,ig(Xm,i, Xm+1,j)|}|Xm,i
]
=
= lim sup
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E
[
exp{α |Ik(Xm,i)Ij(Xm+1,j)|}|Xm,i
]
≤
≤ lim sup
n
|T (n)| − 1
|T (n)|
eα = eα <∞. (48)
Hence it implies that D = Ω. By (7) and (8), we obtain
Fn(ω) =
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
Ik(Xm,i)Il(Xm+1,j) = Sn(k, l), (49)
Gn(ω) =
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E
[
Ik(Xm,i)Il(Xm+1,j)|Xm,i
]
=
=
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
Ik(Xm,i)P (l|Xm,i) =
=
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
Ik(Xm,i)P (l|k) =
= NSn−1(k)P (l|k). (50)
By (49), (50) and (11), noticing lim
n→∞
|T (n)|
/
|T (n−1)| = N, we have
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A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1349
lim
n→∞
1
an
[Fn(ω)−Gn(ω)] =
= lim
n→∞
1
|T (n)|
[
Sn(k, l)−NSn−1(k)P (l|k)
]
=
= lim
n→∞
[
1
|T (n)|
Sn(k, l)− 1
|T (n−1)|
Sn−1(k)P (l|k)
]
= 0. (51)
Hence (47) follows from (51) directly.
Lemma 1 [9]. Let XT (n)
= {Xt, t ∈ T (n)} be a homogeneous Markov chain
indexed by a Cayley tree TC,N which takes values in the finite alphabet set S =
= {1, 2, . . . , N} with the initial distribution p = (p(1), p(2) . . . , p(N)) and transition
matrix (41), assume that the matrix (41) is ergodic. Let Sn(k, ω) be the number of k in
XT (n)
= {Xt, t ∈ T (n)}. Then for all k ∈ S,
lim
n
Sn(k, ω)
|T (n)|
= π(k) µP -a.s., (52)
where π = (π(1), . . . , π(N)) is the stationary distribution determined by P .
Theorem 4. Under the hypothesis of Lemma 1,
lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E
[
exp{α |g(Xm,i, Xm+1,j)|}|Xm,i
]
=
=
∑
k∈S
∑
l∈S
π(k) exp{α|g(k, l)|}P (l|k) µP -a.s. (53)
Proof. By (52) and the definition of Sn(k), we have
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E
[
exp{α |g(Xm,i, Xm+1,j)|}|Xm,i
]
=
=
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
∑
xm+1,j∈S
exp{α|g(Xm,i, xm+1,j)|}P (xm+1,j |Xm,i) =
=
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
∑
l∈S
∑
k∈S
δk(Xm,i) exp{α|g(k, l)|}P (l|k) =
=
∑
l∈S
∑
k∈S
exp{α|g(k, l)|}P (l|k)
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
δk(Xm,i) =
=
∑
l∈S
∑
k∈S
exp{α|g(k, l)|}P (l|k)NSn−1(k). (54)
By (52) and (54), noticing that lim
n→∞
|T (n)|
/
|T (n−1)| = N, we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1350 KANGKANG WANG
lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E
[
exp{α |g(Xm,i, Xm+1,j)|}|Xm,i
]
=
= lim
n
∑
l∈S
∑
k∈S
exp{α|g(k, l)|}P (l|k)
NSn−1(k)
|T (n)|
=
= lim
n
∑
l∈S
∑
k∈S
exp{α|g(k, l)|}P (l|k)
Sn−1(k)
|T (n−1)|
=
=
∑
l∈S
∑
k∈S
π(k) exp{α|g(k, l)|}P (l|k) µP -a.s. (55)
Theorem 5. Let XT (n)
= {Xt, t ∈ T (n)} be a homogeneous Markov chain in-
dexed by a Cayley tree TC,N , we have
lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E [exp{α |g(Xm,i, Xm+1,j)|}] =
=
∑
k∈S
∑
l∈S
p(k) exp{α |g(k, l)|}P (l|k) µP -a.s. (56)
Proof. In virtue of properties of Markov chain, we obtain
lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
E[exp{α |g(Xm,i, Xm+1,j)|}] =
= lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
∑
xm,i∈S
∑
xm+1,j∈S
exp{α |g(xm,i, xm+1,j)|}×
×P (xm,i, xm+1,j) =
= lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
∑
xm,i∈S
∑
xm+1,j∈S
p(xm,i) exp{α |g(xm,i, xm+1,j)|}×
×P (xm+1,j | xm,i) =
= lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
∑
k∈S
∑
l∈S
p(k) exp{α |g(k, l)|}P (l|k) =
=
∑
k∈S
∑
l∈S
lim
n
1
|T (n)|
n−1∑
m=0
Nm∑
i=1
Ni∑
j=N(i−1)+1
p(k) exp{α |g(k, l)|}P (l|k) =
=
∑
k∈S
∑
l∈S
p(k) exp{α |g(k, l)|}P (l|k) µP -a.s. (57)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
A CLASS OF STRONG LIMIT THEOREMS FOR NONHOMOGENEOUS MARKOV CHAINS . . . 1351
Hence (56) follows from (57). We have accomplished the proof.
Acknowledgments. The author would like to thank Professor Weiguo Yang for his
valuable suggestions in the past.
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Received 17.05.09,
after revision — 21.09.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
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| id | umjimathkievua-article-2810 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:30:45Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/04/8528eba54e410afcf7f0f9e5c7503904.pdf |
| spelling | umjimathkievua-article-28102020-03-18T19:37:09Z A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system Про один клас сильних граничних теорем для неоднорiдних марковських ланцюжкiв, що проiндексованi узагальненим деревом бете на узагальненiй системi випадкового вибору Kangkang, Wang Канґканг, Ванг We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree. Вивчаються сильнi граничнi теореми для послiдовностi функцiй двох змiнних неоднорiдного марковського ланцюжка, що проiндексований узагальненим деревом Бете на узагальненiй системi випадкового вибору, шляхом побудови невiд’ємного мартингала. Як наслiдок, узагальнено результати Янга та Є i отримано деякi граничнi теореми для частот станiв, упорядкованих пар та умовного сподiвання функцiї двох змiнних на деревi Келi. Institute of Mathematics, NAS of Ukraine 2011-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2810 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 10 (2011); 1336-1351 Український математичний журнал; Том 63 № 10 (2011); 1336-1351 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2810/2373 https://umj.imath.kiev.ua/index.php/umj/article/view/2810/2374 Copyright (c) 2011 Kangkang Wang |
| spellingShingle | Kangkang, Wang Канґканг, Ванг A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title | A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title_alt | Про один клас сильних граничних теорем для неоднорiдних марковських ланцюжкiв, що проiндексованi узагальненим деревом бете на узагальненiй системi випадкового вибору |
| title_full | A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title_fullStr | A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title_full_unstemmed | A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title_short | A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system |
| title_sort | class of strong limit theorems for nonhomogeneous markov chains indexed by a generalized bethe tree on a generalized random selection system |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2810 |
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