ADVANCES IN MATHEMATICS(CHINA)
doi: 10.11845/sxjz.2012167b
A General Formula on
Tri-point Transition Function With Three States
XIE Shang, XIE Yuquan∗
(School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105,
P. R. China)
Abstract: In this paper, the authors extend an interesting and important result in [Yang
X.Q., Chinese Science Bulletin, 1996, 41(2): 192] and give a general calculating formula on a
homogeneous measurable tri-point transition function with three states.
Key words: two-parameter Markov process; tri-point transition function; three states
MR(2000) Subject Classification: 60J25; 60J35 / CLC number: O211.62
Document code: A
0 Introduction
Tri-point transition function is an important tool to study two-parameter Markov processes and have been minded by many people in probability fields. For example, Cairoli(1971)[1],
Wang(1984)[2], Guo-Yang(1992)[3], Luo(1992)[4], Yang-Li(1996)[5], Bodnariu(2003)[7], Xie(2005,
2006, 2011, 2012)[8−9, 11−12] , etc. In particular, Yang(1996)[6] obtained an interesting and important result and gave a general calculating formula on a homogeneous measurable tri-point
transition function with two states as follows.
Theorem 1[6]
Let E = {0, 1} be a state space. And let P (s, t) = {Pijkr (s, t)} be a
homogeneous measurable tri-point transition function on the state space E. Then P (s, t) is
surely, and only, one of the following three types:

(a)
1 − λ(1 − µt ),



λ(1 − µt ),
Pijkr (s, t) =
(1 − λ)(1 − µt ),



1 − (1 − λ)(1 − µt ),
(i, j) ∈ E,
(i, j) ∈ E,
(i, j) ∈ E,
(i, j) ∈ E,
thus P (s, t) is horizontal constant type;

(b)
1 − λ(1 − µs ),



λ(1 − µs ),
Pijkr (s, t) =
(1 − λ)(1 − µs ),



1 − (1 − λ)(1 − µs ),
(i, k) ∈ E,
(i, k) ∈ E,
(i, k) ∈ E,
(i, k) ∈ E,
k
k
k
k
= 0,
= 0,
= 1,
= 1,
r
r
r
r
= 0,
= 1,
= 0,
= 1,
where λ, µ ∈ [0, 1] are two constant numbers, the right of the formula above is unrelated to s > 0,
j
j
j
j
= 0,
= 0,
= 1,
= 1,
r = 0,
r = 1,
r = 0,
r = 1,
Received date: 2012-11-01. Revised date: 2013-02-18.
Foundation item: This work was supported by Hunan Provincial Science and Technology Department of China(No.
2011FJ6022).
E-mail: ∗ Corresponding author: [email protected]
2
where λ, µ ∈ [0, 1] are two constant numbers, the right of the formula above is unrelated to t > 0,
thus P (s, t) is vertical constant type;
1
(c)
1 + (−1)i−j−k+r pst , i, j, k, r ∈ E,
Pijkr (s, t) =
2
where p ∈ [0, 1] is a constant number, the right of the formula above is only related to st > 0,
thus P (s, t) is parameter symmetric type and is also state symmetric type.
Therefore, it is natural to ask such a question whether there is a similar result as E =
{0, 1, 2}. In order to do this, Xie[9] discovered that a homogeneous measurable tri-point transition
function with three states is surely horizontal constant type or vertical constant type or parameter
symmetric type. However, this result does not give any general calculating formula as Theorem
1. To this end, Xie(2007)[10] again looked back upon the classical transition function with three
states and gave its general calculating formula. In this paper, we will be based on [5, 9–10], and
give a completely satisfactory solution to the above problem.
1 Main Result
Let P (s, t) = {Pijkr (s, t)} be a homogeneous measurable tri-point transition function with
three states. For convenience’s sake, we can assume that the state space E = {0, 1, 2} without
generality.
By [9], we know that P (s, t) is surely horizontal constant type or vertical constant type or
parameter symmetric type. For arbitrary s, t > 0, we define a real valued function

t, P (s, t) is horizontal constant type,
τ = τ (s, t) = s, P (s, t) is vertical constant type,

st, P (s, t) is parameter symmetric type.
Because P (s, t) is Lebesgue measurable, by [5, Chapter 6], all the limits
U = lim P (s, t),
V = lim P (s, t),
τ →+∞
τ →0+
Q = lim
τ →0+
P (s, t) − U
τ
(1)
are existent and U, V are two constant type tri-point transition functions on the state space E.
Base on the essential facts above, we have an important result as follows:
Theorem 2
Let E = {0, 1, 2} be a state space. And let P (s, t) = {Pijkr (s, t)} be a
homogeneous measurable tri-point transition function on the state space E. Then P (s, t) is
surely, and only, one of the following three types:
(a)
P (s, t) = V + (U − V )e−λτ ,
where λ ≥ 0 is a constant number;
(b)
P (s, t) = V + (U − V + τ (Q + λ(U − V )))e−λτ ,
where λ > 0 is a constant number;
β(U − V ) + Q −ατ α(U − V ) + Q −βτ
(c)
e
+
e
,
P (s, t) = V +
β−α
α−β
, : A General Formula on Tri-point Transition Function With Three States
3
where α 6= β are two constant numbers. If α and β are real numbers, then α > β > 0; if not,
√
then α = a + bθ, β = α − bθ, a, b > 0, θ = −1.
In Theorem 2, we give a general calculating formula on a homogeneous measurable tri-point
transition function P (s, t) with three states. It is shown that P (s, t) can be expressed particularly
by some elementary functions on its near limit U , far limit V , density Q and two parameters
s, t > 0. Therefore, the properties of P (s, t) on the two parameters s, t > 0 become very exact. It
is very important for us to study the path properties of a two-parameter Markov process. On the
other hand, if τ and U, V, Q meet some conditions, then the P (s, t) defined by the formulas (a),
(b), (c) is surely a tri-point transition function. For example, if τ and U, V meet the conditions
as follows:
1◦ τ = τ (s, t) = t, ∀t > 0;
2◦ Uijkr ≥ 0, Vijkr ≥ 0, ∀i, j, k, r ∈ E;
P
P
3◦
r∈E Vijkr = 1, ∀i, j, k ∈ E;
r∈E Uijkr =
P
P
◦
4 Uijkr = g∈E Uijlg Ulgkr = g∈E Uilkg Uljgr , ∀i, j, k, r, l ∈ E;
P
P
5◦ Vijkr = g∈E Vijlg Vlgkr = g∈E Vilkg Vljgr , ∀i, j, k, r, l ∈ E;
P
P
6◦ Vijkr = g∈E Uilkg Vljgr = g∈E Vilkg Uljgr , ∀i, j, k, r, l ∈ E;
P
7◦ Uijkr = g∈E Vijlg Ulgkr , ∀i, j, k, r, l ∈ E;
P
8◦ Vijkr = g∈E Uijlg Vlgkr , ∀i, j, k, r, l ∈ E,
then the P (s, t) defined by the formula (a) is surely a horizontal constant type tri-point transition
P
g∈E Vijlg Ulgkr is equal to Uijkr ,
function on the state space E. Especially, in the condition 7◦ ,
not Vijkr .
2 Proof
If P (s, t) is horizontal constant type, then τ = τ (s, t) = t, ∀t > 0. According to the formula
(e) and (f) of [5, Theorem 6.16], we have
Pijkr (s, t) = Pijkr (0, t) =
X X
Ui0kg′ P00g′ g′′ (0, t)U0jg′′ r .
(2)
g′ ∈E g′′ ∈E
where U is defined by the formula (1), that is,
Uijkr = lim Pijkr (0, t),
t→0
∀i, j, k, r ∈ E.
Let
Pkr (t) = P00kr (0, t),
∀t > 0, k, r ∈ E.
(3)
According to the vertical transition equation of [5, Definition 6.1], we have
X
X
Pkr (s + t) = P00kr (0, s + t) =
P00kg (0, s)P00gr (0, t) =
Pkg (s)Pgr (t).
g∈E
g∈E
Therefore P(t) = {Pkr (t), k, r ∈ E} is a homogeneous measurable transition matrix on the state
space E. Let
U = lim P(t),
t→0+
V = lim P(t),
t→+∞
Q = lim
t→0+
P(t) − U
.
t
(4)
4
According to [10, Theorem 1], P(t) is surely, and only, one of the following three types:
(a′ )
P(t) = V + (U − V)e−λt ,
where λ ≥ 0 is a constant number;
(b′ )
P(t) = V + (U − V + t(Q + λ(U − V)))e−λt ,
where λ > 0 is a constant number;
(c′ )
P(t) = V +
β(U − V) + Q −αt α(U − V) + Q −βt
e
+
e ,
β−α
α−β
where α 6= β are two constant numbers. If α and β are real numbers, then α > β > 0; if not,
√
then α = a + bθ, β = α − bθ, a, b > 0, θ = −1.
By (1), (3) and (4), we have
U00kr = Ukr ,
V00kr = Vkr ,
Q00kr = Qkr .
And by (2), we have
Uijkr =
X X
Ui0kg′ U00g′ g′′ U0jg′′ r ,
X X
Ui0kg V00g′ g′′ U0jg′′ r ,
X X
Ui0kg′ Q00g′ g′′ U0jg′′ r .
g′ ∈E
Vijkr =
g′′ ∈E
g′ ∈E g′′ ∈E
Qijkr =
g′ ∈E g′′ ∈E
Therefore, we immediately get the three types (a), (b), (c) in Theorem 2 from (a′ ), (b′ ), (c′ ).
For example, we can obtain (a) from (a′ ) in the way as follows:
X X
Ui0kg′ P00g′ g′′ (0, t)U0jg′′ r
Pijkr (s, t) =
g′ ∈E g′′ ∈E
=
X X
g′ ∈E
=
X X
g′ ∈E g′′ ∈E
=
X X
g′ ∈E g′′ ∈E
=
Ui0kg′ Pg′ g′′ (t)U0jg′′ r
g′′ ∈E
X X
Ui0kg′ Vg′ g′′ + Ug′ g′′ − Vg′ g′′ e−λt U0jg′′ r
Ui0kg′ V00g′ g′′ + U00g′ g′′ − V00g′ g′′ e−λt U0jg′′ r
Ui0kg′ V00g′ g′′ U0jg′′ r
g′ ∈E g′′ ∈E
+
X X
g′ ∈E g′′ ∈E
Ui0kg′ U00g′ g′′ U0jg′′ r −
= Vijkr + (Uijkr − Vijkr )e
−λt
X X
g′ ∈E g′′ ∈E
,
that is,
P (s, t) = V + (U − V )e−λt .
!
Ui0kg′ V00g′ g′′ U0jg′′ r e−λt
, : A General Formula on Tri-point Transition Function With Three States
5
If P (s, t) is horizontal constant type or parameter symmetric type, we can also complete
their proofs in the same way.
Acknowledgements
The authors would like to express their gratitude to editors and
the anonymous referees for their detailed comments and valuable suggestions which considerably
improved the presentation of this paper. This work is a part of Master’s thesis of the first author.
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