A limit theorem for semi-Markov process
A limit theorem for the strongly regular semi-Markov process is proved under conditions C1 – C3.
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| Цитувати: | A limit theorem for semi-Markov process / A. Bondarenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 35-43. — Бібліогр.: 8 назв.— англ. |
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irk-123456789-44762009-11-20T12:00:55Z A limit theorem for semi-Markov process Bondarenko, A. A limit theorem for the strongly regular semi-Markov process is proved under conditions C1 – C3. 2007 Article A limit theorem for semi-Markov process / A. Bondarenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 35-43. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4476 en Інститут математики НАН України |
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A limit theorem for the strongly regular semi-Markov process is proved under conditions C1 – C3. |
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Bondarenko, A. |
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Bondarenko, A. A limit theorem for semi-Markov process |
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Bondarenko, A. |
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Bondarenko, A. |
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A limit theorem for semi-Markov process |
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A limit theorem for semi-Markov process |
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A limit theorem for semi-Markov process |
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A limit theorem for semi-Markov process |
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A limit theorem for semi-Markov process |
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limit theorem for semi-markov process |
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Інститут математики НАН України |
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2007 |
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A limit theorem for semi-Markov process / A. Bondarenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 35-43. — Бібліогр.: 8 назв.— англ. |
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AT bondarenkoa alimittheoremforsemimarkovprocess AT bondarenkoa limittheoremforsemimarkovprocess |
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2025-07-02T07:42:44Z |
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2025-07-02T07:42:44Z |
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1836520219262058496 |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.35-43
ANNA BONDARENKO
A LIMIT THEOREM FOR SEMI-MARKOV
PROCESS
A limit theorem for the strongly regular semi-Markov process is
proved under conditions C1 – C3.
1. Introduction
This article deals with the asymptotic behavior of the strongly regular
semi-Markov process ξ(t) as t → ∞. It may be considered as continuation
of the article [1] motivated by the book by A. N. Korlat, V. N. Kuznetsov,
M. M. Novikov and A. F. Turbin (1991). Let us introduce basic notations
and necessary results from [1], [2].
Let ξ(t) be a strongly regular semi-Markov process with the phase space
{X,B} and semi-Markov kernel Q(t, x, B), t ≥ 0, x ∈ X, B ∈ B. Let
H(t, x, B), t ≥ 0, x ∈ X, B ∈B be the Markov renewal function of ξ(t).
Define D(X) as Banach space of B - measurable bounded functions with
values in R with the norm ‖f‖ = supx∈X |f(x)|. Consider two operator
family Q(t) and H(t), t ≥ 0, in D(X), defined for all f ∈ D(X):
[Q(t)f ](x) =
∫
X
Q(t, x, dy)f(y),
[H(t)f ](x) =
∫
X
H(t, x, dy)f(y).
Suppose that ξ(t) satisfies the following conditions:
C1. Markov chain ξn, n ≥ 0, embedded in the ξ(t), is uniformly recurring;
C2. ‖Ml‖ < ∞ for l = 1, k + 2, k ≥ 1, where Ml =
∞∫
0
tlQ(dt);
C3. Semi-Markov kernel of the process ξ(t) is absolutely continuous in t:
Q(t, x, B) =
t∫
0
q(s, x, B)ds, t ≥ 0, x ∈ X, B∈B,
2000 Mathematics Subject Classifications: 60K15, 60K20
Key words and phrases. Strongly regular semi-Markov process, embedded Markov
chain, uniformly recurring, Markov renewal theorem.
35
36 ANNA BONDARENKO
or in the operator form:
Q(t) =
t∫
0
q(s)ds, t ≥ 0.
Condition C3 guarantees existence of the density of the Markov renewal
function h(t, x, B):
H(t, x, B) = IB(x) +
t∫
0
h(s, x, B)ds, t ≥ 0, x ∈ X, B∈B,
or in the operator form
H(t) = I +
t∫
0
h(s)ds, t ≥ 0,
where I is the identity operator, IB(x) is the indicator function.
Let Π0 be the stationary projector of the embedded Markov chain ξn
defined under condition C1 as follows:
[Π0f ](x) =
∫
X
ρ(dy)f(y)I(x), ∀f ∈ D(X)
where ρ(x) is the stationary distribution of the Markov chain ξn, I(x) ≡ 1
∀x ∈ X. Denote
h∗(t) = h(t) − 1
m̂1
Π0, (1)
where
m̂1 =
∫
X
ρ(dx)m1(x), m1(x) =
∞∫
0
tQ(dt, x, X).
Let Tn, n = 0, k be bounded operators in D(X), introduced in the book
[2, p. 1.4], and let P = Q(∞) be the operator of transient probabilities of
Markov chain ξn. The following result was proved for n = 0 in [2] and for
n = 1, k in [1]:
Theorem 1. Let a strongly regular semi-Markov process satisfies conditions
C1 – C3. Then there exists the limit
Un = lim
p→0
(−1)n
n!
∞∫
0
e−pttnh∗(t)dt, n = 0, k (2)
LIMIT THEOREM FOR SAMI-MARKOV PROCESS 37
and the following relations hold:
Un =
n∑
r=0
(−1)r
(r)!
MrUn−r +
(−1)n
n!
Mn +
(−1)n+1
(n + 1)!m̂1
Mn+1Π0, n = 0, k, (3)
Un =
{
T0 − I, for n = 0;
Tn, for n = 1, k,
(4)
where M0 = P .
2. Basic results.
In this paper we present a theorem, which is proved by means of the
above mentioned results and the Markov renewal theorem.
Let’s introduce a family of operators
U0(t) =
t∫
0
h∗(s)ds, Un(t) =
t∫
0
(Un−1(s)−Un−1)ds, t ≥ 0, n = 1, k. (5)
The following result holds true:
Theorem 2. Let a strongly regular semi-Markov process satisfies conditions
C1 – C3. Then there exists the limit
lim
t→∞
Un(t) = Un, t ≥ 0, n = 0, k. (6)
Proof. 1. Consider the case n = 0. Under condition C3 the operator
renewal equation holds true [3]:
h(t) = q(t) +
t∫
0
q(s)h(t − s)ds.
Hence, subject to (1)
h∗(t) = q(t) − 1
m̂1
(I − Q(t))Π0 +
t∫
0
q(s)h∗(t − s)ds. (7)
Taking integral of (7) and using the Fubbini theorem [4] we will get
t∫
0
h∗(s)ds = Q(t) − 1
m̂1
t∫
0
(I − Q(s))ds Π0 +
t∫
0
ds q(s)
t−s∫
0
h∗(l)dl,
38 ANNA BONDARENKO
or
U0(t) = Q(t) − 1
m̂1
t∫
0
(I − Q(s))ds Π0 +
t∫
0
q(s)U0(t − s)ds. (8)
In the case n = 0 from (3) we get
U0 = P + PU0 − 1
m̂1
M1Π0. (9)
Taking into account the property of stationary projector Π0 :
PΠ0 = Π0 = Π0P, (10)
consider the difference between (8) and (9):
U0(t) − U0 = V0(t) +
t∫
0
q(s)(U0(t − s) − U0)ds, (11)
where
V0(t) =
∞∫
t
(P − Q(s))ds
Π0
m̂1
+ Q(t) − P − (P − Q(t))U0. (12)
Lemma 1. Let conditions of Theorem 2 be satisfied. Then there exists the
limit
lim
t→∞
(U0(t) − U0) =
Π0
m̂1
∞∫
0
V0(s)ds. (13)
Proof. To prove the operator equation (13) it is sufficient to verify it for
functions IB(x), x ∈ X, B ∈ B, generating D(X). Define V0(t, x, B),
U0(t, x, B), U0(x, B) as action of operators V0(t), U0(t), U0 on function
IB(x). Consider positive and negative parts of the function V0(t, x, B):
V 1
0 (t, x, B) := max{V0(t, x, B), 0}, V 2
0 (t, x, B) := −min{V0(t, x, B), 0}.
Similarly U1
0 (x, B) and U2
0 (x, B) are defined as positive and negative parts
of function U0(x, B). From (12) it follows that for t ≥ 0, x ∈ X, B ∈ B
V 1
0 (t, x, B) =
ρ(B)
m̂1
∞∫
t
dt0
∞∫
t0
q(s, x, X)ds +
∞∫
t
ds
∫
X
q(s, x, dy)U2
0 (y, B),
V 2
0 (t, x, B) =
∞∫
t
q(s, x, B)ds +
∞∫
t
ds
∫
X
q(s, x, dy)U1
0 (y, B).
LIMIT THEOREM FOR SAMI-MARKOV PROCESS 39
Functions V 1
0 (t, x, B) and V 2
0 (t, x, B) are bounded. It follows from condition
C2 for l = 1 and boundedness of the operator T0 = U0. Besides, for any
x ∈ X, B∈B functions V 1
0 (t, x, B), V 2
0 (t, x, B) are non-negative, monotone
decreasing and integrable in t functions on [0,∞). Thus for any B ∈ B
they are directly Riemann integrable [5], so that
∫
X
ρ(dx)
∞∫
0
dtV j
0 (t, x, B) <
∞, j = 1, 2. So for a fixed B∈B the above point and conditions C1 – C3
give a possibility to apply the Markov renewal theorem ([5, p. 107], [6, p.
31] ) to the following Markov renewal equation:
Zj(t, x, B) = V j
0 (t, x, B) +
t∫
0
ds
∫
X
q(s, x, dy)Zj(t− s, y, B), j = 1, 2. (14)
By the Markov renewal theorem there exists
lim
t→∞
Zj(t, x, B) =
1
m̂1
∫
X
ρ(dx)
∞∫
0
dtV j
0 (t, x, B), x ∈ X, B ∈ B. (15)
As by definition V 1
0 (t, x, B) − V 2
0 (t, x, B) = V0(t, x, B), then from (11) and
(14) it follows that U0(t, x, B)−U0(x, B) = Z1(t, x, B)−Z2(t, x, B). Hence,
from (15) follows statement of the lemma.
Since ∫ ∞
0
V0(s)ds = −M1 − M1U0 +
M2Π0
2m̂1
,
then from (3) for n = 1 and Lemma 1 we get
lim
t→∞
(U0(t) − U0) =
Π0
m̂1
(I − P )U1 = 0.
Theorem 2 for n = 0 is proved.
2. Consider the case n = 1, k. Define
E0(t) =
∞∫
t
dt0
∞∫
t0
q(s)ds, E1(t) =
∞∫
t
dt1
∞∫
t1
dt0
∞∫
t0
q(s)ds,
En(t) =
∞∫
t
dtn
∞∫
tn
dtn−1...
∞∫
t1
dt0
∞∫
t0
q(s)ds, n = 2, k.
It is easy to see that for t = 0 the following equalities hold true:
E0(0) =
∞∫
0
(P − Q(t))dt = M1,
40 ANNA BONDARENKO
En(0) =
∞∫
0
En−1(t)dt =
Mn+1
(n + 1)!
, n = 1, k + 1. (16)
Transform (3) to the form
Un =
n−1∑
r=0
(−1)(r+1)Er(0)Un−r−1+
+(−1)(n+1)En(0)
Π0
m̂1
+ PUn + (−1)nEn−1(0). (17)
Lemma 2. Let conditions of Theorem 2 be satisfied. Then the following
relations hold true:
Un(t) − Un = Vn(t) +
t∫
0
q(s)(Un(t − s) − Un)ds, t ≥ 0, n = 1, k, (18)
where
Vn(t) =
n−1∑
r=0
(−1)rEr(t)Un−r−1+
+(−1)nEn(t)
Π0
m̂1
+ (−1)n−1En−1(t) −
∞∫
t
q(s)dsUn. (19)
Proof. The lemma is proved by means of mathematical induction method.
From (5) and (11) we have
U1(t) = −E0(t)U0 + E1(t)
Π0
m̂1
− E0(t) +
t∫
0
q(s)U1(t − s)ds. (20)
From (17) for n = 1 and (20) we obtain statement of the lemma for the
case n = 1. So we have the base of induction. Suppose that statement of
the lemma is true for some n, n = 1, k − 1 and show that it is also true for
n + 1. Indeed, let us integrate (18) and apply the Fubbini theorem. We get
Un+1(t) =
t∫
0
Vn(s)ds +
t∫
0
ds q(s)
t−s∫
0
(Un(l) − Un)dl ±
∞∫
0
Vn(s)ds,
or
Un+1(t) = −
∞∫
t
Vn(s)ds +
∞∫
0
Vn(s)ds +
t∫
0
q(s)Un+1(t − s)ds. (21)
LIMIT THEOREM FOR SAMI-MARKOV PROCESS 41
Since by definition
∞∫
t
En(s)ds = En+1(t), we have
∞∫
t
Vn(s)ds =
n−1∑
r=0
(−1)rEr+1(t)Un−r−1+
+(−1)nEn+1(t)
Π0
m̂1
+ (−1)n−1En(t) − E0(t)Un =
=
n∑
r=0
(−1)(r+1)Er(t)Un−r + (−1)nEn+1(t)
Π0
m̂1
+ (−1)n−1En(t). (22)
It follows from (22) for t = 0 and (17) that
∞∫
0
Vn(s)ds = Un+1 − PUn+1. (23)
So if relation (18) is true for some n, n = 1, k − 1, then, as follows from
(21), (22) and (23), it is also true for n + 1.
Prove the next lemma in the way similar to one of Lemma 1.
Lemma 3. Let conditions of Theorem 2 are satisfied. Then there exists the
limit
lim
t→∞
(Un(t) − Un) =
Π0
m̂1
∞∫
0
Vn(s)ds, n = 1, k. (24)
Proof. To prove the operator equation (24) it is sufficient to verify it for
the indicator functions IB(x), x ∈ X, B ∈ B, generating D(X). Define
Vn(t, x, B) and Ur(x, B), r = 0, n as action of operators Vn(t), Ur on
function IB(x). From (19) it follows
Vn(t, x, B) = (−1)n ρ(B)
m̂1
∞∫
t
dtn
∞∫
tn
dtn−1...
∞∫
t1
dt0
∞∫
t0
q(s, x, X)ds+ (25)
+
n−1∑
r=0
(−1)r
∞∫
t
dtr
∞∫
tr
dtr−1...
∞∫
t1
dt0
∞∫
t0
ds
∫
X
q(s, x, dy)Un−r−1(y, B)+
+(−1)n−1
∞∫
t
dtn−1...
∞∫
t1
dt0
∞∫
t0
q(s, x, B)ds −
∞∫
t
ds
∫
X
q(s, x, dy)Un(y, B).
42 ANNA BONDARENKO
Consider positive and negative parts of the function Vn(t, x, B):
V 1
n (t, x, B) := max{Vn(t, x, B), 0}, V 2
n (t, x, B) := −min{Vn(t, x, B), 0}.
Represent functions Ur(x, B), r = 0, n in (25) as U1
r (x, B)−U2
r (x, B) where
U1
r (x, B) and U2
r (x, B) are its positive and negative parts. Then from (25)
it follows that Vn(t, x, B) is a sum of functions of constant signs. It is easy
to see from (25) the structure of functions V +
n and V −
n and make a conclu-
sion, that for any fixed x ∈ X, B∈B functions V +
0 (t, x, B), V −
0 (t, x, B) are
non-negative, monotone decreasing and integrable in t functions on [0,∞).
Boundedness of this functions follows from the condition C2, (4) and bound-
edness of operators Ti, i = 0, n. Thus V 1
n and V 2
n are directly Riemann
integrable, so that
∫
X
ρ(dx)
∞∫
0
dtV j
n (t, x, B) < ∞, j = 1, 2. Hence the above
point and conditions C1 – C3 give a possibility to apply the Markov renewal
theorem to the next equation:
Zj(t, x, B) = V j
n (t, x, B) +
t∫
0
ds
∫
X
q(s, x, dy)Zj(t− s, y, B), j = 1, 2. (26)
By the Markov renewal theorem there exists
lim
t→∞
Zj(t, x, B) =
1
m̂1
∫
X
ρ(dx)
∞∫
0
dtV j
n (t, x, B), x ∈ X, B ∈ B. (27)
Note that by definition V 1
n (t, x, B) − V 2
n (t, x, B) = Vn(t, x, B). So from
(26) and Lemma 2 it follows that Un(t, x, B) − Un(x, B) = Z1(t, x, B) −
Z2(t, x, B). From (27) statement of the lemma follows.
From (23) and (10) it follows that
Π0
∞∫
0
Vn(s)ds = Π0(Un+1 − PUn+1) = 0.
Using Lemma 3, we get statement of the theorem for n = 1, k.
Conclusion.
In Theorem 2 the asymptotic equality (6) is proved for a strongly regu-
lar semi-Markov process which satisfies conditions C1 – C3. In the case
n = 0 this asymptotic equality follows from results of [2] but under two
LIMIT THEOREM FOR SAMI-MARKOV PROCESS 43
additional conditions. Note, that (6) is more weak result than existence of∫ ∞
0
tnh∗(t)dt, n = 0, k. Indeed, if such integral exists, then
Un =
(−1)n
n!
∞∫
0
tnh∗(t)dt, n = 0, k,
and according to formula of integration by parts, we get
∞∫
0
tnh∗(t)dt = n!
∞∫
0
dtn
∞∫
tn
dtn−1...
∞∫
t2
dt1
∞∫
t1
h∗(t)dt, n = 1, k.
from which asymptotic equality (6) follows. However, as far as the author
knows, at the present moment existence of
∫ ∞
0
tnh∗(t)dt for the general
semi-Markov process is not proved. It is known that such integral is con-
vergent for the renewal process under conditions that are a particular case
of the conditions C1-C3 for the renewal process ([7], [8]).
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: gannucia@ukr.net
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