Investigation of the asymptotics of a renewal matrix
A semi-Markov process with finite state space and continuous time without finiteness condition of a mean stay time of this process in every fixed state is considered. The asymptotic behaviour of the renewal matrix at infinity is established under condition that the distribution tail of the stay time of...
Saved in:
| Date: | 2006 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2006
|
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/4439 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Investigation of the asymptotics of a renewal matrix / N. V. Buhrii // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 33–37. — Бібліогр.: 8 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4439 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-44392009-11-11T12:00:28Z Investigation of the asymptotics of a renewal matrix Buhrii, N.V. A semi-Markov process with finite state space and continuous time without finiteness condition of a mean stay time of this process in every fixed state is considered. The asymptotic behaviour of the renewal matrix at infinity is established under condition that the distribution tail of the stay time of the semi-Markov process in every fixed state is a regularly varying function at infinity with exponent −1. 2006 Article Investigation of the asymptotics of a renewal matrix / N. V. Buhrii // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 33–37. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4439 519.21 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A semi-Markov process with finite state space and continuous time without finiteness
condition of a mean stay time of this process in every fixed state is considered. The asymptotic behaviour of the renewal matrix at infinity is established under condition that the distribution tail of the stay time of the semi-Markov process in every fixed state is a regularly varying function at infinity with exponent −1. |
| format |
Article |
| author |
Buhrii, N.V. |
| spellingShingle |
Buhrii, N.V. Investigation of the asymptotics of a renewal matrix |
| author_facet |
Buhrii, N.V. |
| author_sort |
Buhrii, N.V. |
| title |
Investigation of the asymptotics of a renewal matrix |
| title_short |
Investigation of the asymptotics of a renewal matrix |
| title_full |
Investigation of the asymptotics of a renewal matrix |
| title_fullStr |
Investigation of the asymptotics of a renewal matrix |
| title_full_unstemmed |
Investigation of the asymptotics of a renewal matrix |
| title_sort |
investigation of the asymptotics of a renewal matrix |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4439 |
| citation_txt |
Investigation of the asymptotics of a renewal matrix / N. V. Buhrii // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 33–37. — Бібліогр.: 8 назв.— англ. |
| work_keys_str_mv |
AT buhriinv investigationoftheasymptoticsofarenewalmatrix |
| first_indexed |
2025-07-02T07:41:03Z |
| last_indexed |
2025-07-02T07:41:03Z |
| _version_ |
1836520113206984704 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 33–37
UDC 519.21
N. V. BUHRII
INVESTIGATION OF THE ASYMPTOTICS
OF A RENEWAL MATRIX
A semi-Markov process with finite state space and continuous time without finiteness
condition of a mean stay time of this process in every fixed state is considered. The
asymptotic behaviour of the renewal matrix at infinity is established under condition
that the distribution tail of the stay time of the semi-Markov process in every fixed
state is a regularly varying function at infinity with exponent −1.
Introducton
The classical renewal theorems give the asymptotic behaviour of the renewal function
associated with a distribution F at infinity (see [1, §12.1]). These theorems are valid when
the mean of the distribution F is equal to infinity. In the case of a regularly varying
distribution function tail 1−F at infinity with exponent −α, 0 ≤ α ≤ 1, Erickson defined
more precisely the behaviour of the renewal function at infinity (see [2, Theorems 1,5]).
An analogy of the renewal function for a semi-Markov process with finite state space
is the so-called renewal matrix associated with a semi-Markov matrix P . If the matrix
P (∞) is irreducible and a mean stay time of the semi-Markov process in every fixed
state is finite, then the asymptotic behaviour of the renewal matrix is well known (see
[1, §12.6]). V.S. Korolyuk and A.F. Turbin have investigated a limit properties of the
renewal matrix too (see, [3, Chapter 4]). In this paper, a semi-Markov process with finite
state space without finiteness condition of a mean stay time of this process in every fixed
state is considered. The asymptotics of the renewal matrix is found under condition
that the distribution tail of this stay time is a regularly varying function at infinity with
exponent α = −1.
The asymptotic behaviour of the renewal matrix
Let X(t), t ≥ 0, be a semi-Markov process with finite state space {1, 2, . . . , m} and
continuous time. Define
τ1 = τ = inf{t > 0 : X(t) �= X(0)}, . . . ,
τn = inf{t > τn−1 : X(t) �= X(τn−1)}, n ≥ 2.
A sequence of the random quantities X(0), X(τ1), . . . , X(τn), . . . creates the so-called
imbedded Markov chain in X(t) with the transition probabilities
pij = P{X(τ) = j | X(0) = i}, i, j = 1, m.
2000 AMS Mathematics Subject Classification. Primary 60J25.
Key words and phrases. Semi-Markov process, renewal matrix, regular and slow variation, infinite
mean.
33
34 N. V. BUHRII
The matrix P = ||pij ||mi,j=1 is irreducible. Therefore, a unique stationary distribution p1,
p2, . . . , pm exists such that
pi ≥ 0,
m∑
i=1
pi = 1, pj =
m∑
i=1
pipij , j = 1, m.
Denote
Fij(t) = P{τ ≤ t, X(τ) = j | X(0) = i},
Fi(t) =
m∑
j=1
Fij(t) = P{τ ≤ t | X(0) = i}, i = 1, m.
Assume that the mean stay time of the semi-Markov process X(t) in every fixed state is
infinite, that is,
Miτ = M{τ | X(0) = i} =
∫ ∞
0
x dFi(x) =
∫ ∞
0
(1 − Fi(x)) dx = +∞,
i = 1, m.
Consider the Markov renewal equation (see [3, p. 38])
qi(t) = bi(t) +
m∑
j=1
∫ t
0
Fij{dx}qj(t − x), i = 1, m, t ≥ 0, (1)
where the matrix F (t) = ||Fij(t)||mi,j=1 is semi-Markov.
The solution of Eq. (1) is (see [3, p. 40])
qi(t) =
m∑
j=1
∫ t
0
Uij{dx}bj(t − x) =
=
m∑
j=1
∫ 1
0
Uij{tdx}bj(t(1 − x)), i = 1, m, (2)
where Uij{[0, t]} = Uij(t), i, j = 1, m, are the elements of the renewal matrix U(t) which
is an analogy of the renewal function, that is,
U(t) =
∞∑
n=0
Fn∗(t), (3)
where F 0∗(t) = E ≡ ||δij ||mi,j=1 is the unit matrix,
F 1∗(t) = F (t),
F (n+1)∗(t) =
∫ t
0
Fn∗(t − u)F{du}, n ∈ N.
Theorem 1. Let
1 − Fi(t) ∼ ai
L(t)
t
, i = 1, m, as t → ∞, (4)
where L is a slowly varying function at infinity, and a1, . . . , am are some nonnegative
constants such that
∑m
i=1 aipi > 0. Then
lim
t→∞
1
t
L1(t)Uij(t) =
pj∑m
i=1 aipi
, i, j = 1, m, (5)
where
L1(t) =
∫ t
1/d
L(x)
x
dx, d > 0.
INVESTIGATION OF THE ASYMPTOTICS OF A RENEWAL MATRIX 35
Remark 1. Since
∑m
i=1 pi = 1, the condition
∑m
i=1 aipi > 0 excludes a possibility,
when ai = 0, i = 1, m.
Proof. Find the asymptotics of the renewal measure U(tx) = ||Uij(tx)||mi,j=1, x ∈ [0, 1],
as t → ∞. Denote, by
Û(λ) =
∫ ∞
0
e−λxU{dx}, λ > 0,
the Laplace transform of a measure U and, by
F̂ (λ) =
∫ ∞
0
e−λxF{dx}, λ > 0,
the Laplace transform of a distribution F .
From (3), using properties of the Laplace transformation (see [4, p. 500]), we have
Û(λ) =
∞∑
n=0
(
F̂ (λ)
)n
.
Count the norm of the matrix F̂ (λ):
||F̂ (λ)|| = sup
1≤i≤m
m∑
j=1
|F̂ij(λ)| = sup
1≤i≤m
m∑
j=1
∫ ∞
0
e−λxFij{dx} =
= sup
1≤i≤m
∫ ∞
0
e−λxFi{dx}.
Integrating by parts, we obtain∫ ∞
0
e−λxFi{dx} = e−λxFi(x)
∣∣∣∣
∞
0
+λ
∫ ∞
0
Fi(x)e−λxdx =
∫ ∞
0
Fi
(
t
λ
)
e−tdt.
By the geometrical matter of an integral, we get∫ ∞
0
Fi
(
t
λ
)
e−tdt <
∫ ∞
0
e−tdt, i = 1, m.
Therefore
||F̂ (λ)|| = sup
1≤i≤m
∫ ∞
0
Fi
(
t
λ
)
e−tdt < 1.
Thus, by Theorem 5 in [5, p. 216], we obtain
∞∑
n=0
(
F̂ (λ)
)n
= [E − F̂ (λ)]−1.
Hence,
Û(λ) = [E − F̂ (λ)]−1. (6)
For a fixed λ, consider the matrix sequence
{
F̂
(
λ
t
)}
as t → ∞. It is a monotonous
nondecreasing matrix sequence with nonnegative elements, for which
F̂
(
λ
t
)
→ P for t → ∞,
where P = ||pij ||mi,j=1 is the transition matrix of the imbedded Markov chain. Since P is
an irreducible matrix, we obtain (see [6, p. 599])
ct
(
E − F̂
(
λ
t
))−1
→ ||1 · pj||mi,j=1 as t → ∞, (7)
36 N. V. BUHRII
where ct = 1 −
(
p, F̂
(
λ
t
)
· 1
)
, p = (p1, . . . , pm), 1 = (1, . . . , 1).
It follows from Theorem 1 in [7, p. 30] that, for all λ > 0,
1 − F̂i
(
λ
t
)
∼ λai
t
· L1
(
t
λ
)
, i = 1, m, as t → ∞.
Consequently,
ct = 1 −
(
p, F̂
(
λ
t
)
· 1
)
=
m∑
i=1
pi
(
1 − F̂i
(
λ
t
))
∼
∼
m∑
i=1
pi
λai
t
· L1
(
t
λ
)
as t → ∞.
Since L1 is a slowly varying function at infinity (see [8, p. 220]), we get
ct ∼ λ
t
L1(t)
m∑
i=1
aipi as t → ∞. (8)
Thus, in view of (7) and (8), relation (7) yields
λ
t
L1(t)
m∑
i=1
aipi
∫ ∞
0
e−
λ
t xU{dx} → ||1 · pj ||mi,j=1 as t → ∞,
that is,
1
t
L1(t)
∫ ∞
0
e−λuU{tdu} → 1
λ
∑m
i=1 aipi
· ||1 · pj ||mi,j=1 as t → ∞
Whence we get
1
t
L1(t)
∫ ∞
0
e−λuUij{tdu} → pj
λ
∑m
i=1 aipi
, i, j = 1, m, as t → ∞.
By the generalized continuity theorem (see [4, c. 499]), the quantity
pj
λ
∑m
i=1 aipi
is a Laplace transform of some measure pjμ1 such that∫ ∞
0
e−λuμ1{du} =
1
λ
∑m
i=1 aipi
, λ > 0, (9)
and
1
t
L1(t)Uij{tdu} → pjμ1{du}, i, j = 1, m, as t → ∞ (10)
for every bounded continuity interval of the measure μ1.
Since ∫ ∞
0
e−λudu =
1
λ
,
we see from (9) that μ1 is the Lebesgue measure multiplied by a constant. Therefore,
the interval (0, 1) is a continuity interval of the measure μ1. Thus, we get from (10) that
lim
t→∞
1
t
L1(t)Uij{(0, t)} = pjμ1{(0, 1)} =
pj∑m
i=1 aipi
, i, j = 1, m.
INVESTIGATION OF THE ASYMPTOTICS OF A RENEWAL MATRIX 37
Note that Uij{(0, t)} = Uij(t− 0)−Uij(0), i, j = 1, m. From (3), we obtain Uij(0) = δij ,
i, j = 1, m. Since L1(t) → ∞ as t → ∞, we count, by the L’Hospital rule,
lim
t→∞
L1(t)
t
= lim
t→∞
∫ t
1/d
L(x)
x
dx
t
= lim
t→∞
L(t)
t
= 0,
i = 1, m. Consequently,
lim
t→∞
1
t
L1(t)Uij{(0, t)} = lim
t→∞
1
t
L1(t)[Uij(t − 0) − δij ] =
= lim
t→∞
1
t
L1(t)Uij(t) =
pj∑m
i=1 aipi
, i, j = 1, m.
Theorem 1 is proved.
Bibliography
1. I.N. Kovalenko, N.Yu. Kuznetsov, V.M. Shurenkov, Chance Processes: Reference Book, Nauko-
va Dumka, Kyiv, 1983.
2. K.B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc. 151
(1970), 263–291.
3. V.S. Korolyuk, A.F. Turbin, Semi-Markov Processes and Their Applications, Naukova Dumka,
Kyiv, 1976.
4. W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New York, 1970.
5. A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis,
Nauka, Moscow, 1972.
6. V.M. Shurenkov, Ya.I. Yeleiko, Limit distributions of time mean values for a semi-Markov
process with finite stste space, Ukr. Mat. Zh. 31 (1979), no. 5, 598–603.
7. Ya.I. Yeleiko, N.V. Buhrii, An asymptotics of Laplace transform of a distribution which has
a regularly varying tail with exponent -1, Math. Meth. Physicomech. Fields 44 (2001), no. 2,
30–33.
8. S. Parameswaran, Partition function whose logarithms are slowly oscillating, Trans. Amer. Soc.
100 (1961), 217–240.
E-mail : ol buhrii@ua.fm
|