Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes
Asymptotic expansions are given for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes.
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| Цитувати: | Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 267-271. — Бібліогр.: 1 назв.— англ. |
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irk-123456789-44952009-11-20T12:00:36Z Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes Silvestrov, D.S. Asymptotic expansions are given for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes. 2007 Article Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 267-271. — Бібліогр.: 1 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4495 en Інститут математики НАН України |
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Asymptotic expansions are given for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes. |
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Article |
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Silvestrov, D.S. |
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Silvestrov, D.S. Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
| author_facet |
Silvestrov, D.S. |
| author_sort |
Silvestrov, D.S. |
| title |
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
| title_short |
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
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Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
| title_fullStr |
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
| title_full_unstemmed |
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes |
| title_sort |
asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-markov processes |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4495 |
| citation_txt |
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 267-271. — Бібліогр.: 1 назв.— англ. |
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2025-07-02T07:43:35Z |
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2025-07-02T07:43:35Z |
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1836520272891478016 |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.267-271
DMITRII S. SILVESTROV
ASYMPTOTIC EXPANSIONS FOR
QUASI-STATIONARY DISTRIBUTIONS
OF NONLINEARLY PERTURBED
SEMI-MARKOV PROCESSES
Asymptotic expansions are given for quasi-stationary distributions
of nonlinearly perturbed semi-Markov processes.
1. Introduction
Quasi-stationary phenomena in stochastic systems are a subject of in-
tensive studies that started in 60s of the 20th century. These phenomena
describe the behaviour of stochastic systems with random lifetimes. The
core of the quasi-stationary phenomenon is that one can observe something
that resembles a stationary behaviour of the system before the lifetime goes
to the end.
Examples of stochastic systems, in which quasi-stationary phenomena
can be observed, are various queuing systems and reliability models, in
which the lifetime is usually considered to be the time in which some kind
of a fatal failure occurs in the system. Another class of examples of such
stochastic systems is supplied by population dynamics or epidemic models.
In population dynamics models, the lifetimes are usually the extinction
times for the corresponding populations. In epidemic models, the role of the
lifetime is played by the time of extinction of the epidemic in the population.
Usually the behaviour of a stochastic system can be described in terms of
some Markov type stochastic process η(ε)(t) and its lifetime defined to be the
time μ(ε) at which the process η(ε)(t) hits a special absorption subset of the
phase space of this process for the first time. A typical situation is when the
process η(ε)(t) and the absorption time μ(ε) depend on a small parameter ε ≥
0 in the sense that some of their local “transition” characteristics depend
Invited lecture
2000 Mathematics Subject Classifications: 60K15, 60F17, 60K20.
Key words and phrases: semi-Markow processes, absorbtion time, quasi-stationary
distribution, nonlinear perturbation.
267
268 DMITRII S. SILVESTROV
on the parameter ε. The parameter ε is involved in the model in such a
way that the corresponding local characteristics are continuous at the point
ε = 0, if regarded as functions of ε. These continuity conditions permit to
consider the process η(ε)(t), for ε > 0 as a perturbed version of the process
η(0)(t). In many cases, under some natural communication and aperiodicity
conditions imposed on the local transition characteristics of the process
η(ε)(t), there exists a so-called quasi-stationary distribution for the process
η(ε)(t), which is given by the formula,
π(ε)(A) = lim
t→∞
P{η(ε)(t) ∈ A / μ(ε) > t}. (1)
In this paper, we give asymptotic expansions for quasi-stationary dis-
tributions of nonlinearly perturbed semi-Markov processes. The principal
novelty of the result is that we consider the model with nonlinear pertur-
bations. By a nonlinear perturbation we mean that local transition char-
acteristics (that are some moment functionals of transition probabilities for
the corresponding semi-Markov processes) are nonlinear functions of the
perturbation parameter ε and that the assumptions made imply that the
characteristics can be expanded in an asymptotic power series with respect
to ε up to and including some order k.
2. Main result
Let, for every ε ≥ 0, (η
(ε)
n , κ
(ε)
n ), n = 0, 1, . . ., be a Markov renewal process
with the phase space X = {0, 1, . . . , N} and transition probabilities Q
(ε)
ij (u).
Let ν(ε)(t) = max(n : τ (ε)(n) ≤ t), t ≥ 0, where 0 = τ (ε)(0), τ (ε)(n) =
κ
(ε)
1 + . . . κ
(ε)
n , n ≥ 1, and η(ε)(t) = η
(ε)
ν(ε)(t)
, t ≥ 0, be the corresponding semi-
Markov process associated with the Markov renewal process (η
(ε)
n , κ
(ε)
n ).
We write the transition probabilities as Q
(ε)
ij (t) = p
(ε)
ij F
(ε)
ij (t), where
p
(ε)
ij = Q
(ε)
ij (∞) are the transition probabilities of the corresponding imbed-
ded Markov chain and F
(ε)
ij (t) are the distribution functions of transition
times. For simplicity we assume that the following condition preventing
instant jumps holds:
I: F
(ε)
ij (0) = 0, i, j ∈ X for all ε ≥ 0.
We consider a model in which 0 is an absorbing state, that is, the fol-
lowing condition holds:
A: p
(ε)
0j = 0, j �= 0, for all ε ≥ 0.
Let also ν
(ε)
j = min{n ≥ 1 : η
(ε)
n = j} and μ
(ε)
j = τ (ε)(ν
(ε)
j ) be the first
hitting time at which the imbedded Markov chain η
(ε)
n and the semi-Markov
NONLINEARLY PERTURBED SEMI-MARKOV PROCESSES 269
process η(ε)(t) hit the state j, respectively. The first hitting time μ
(ε)
0 is the
absorption time for the semi-Markov process η(ε)(t).
We also assume the following condition imposed on the transition prob-
abilities of the semi-Markov process η(ε)(t), which allows to consider these
processes for ε > 0 as perturbed versions of the semi-Markov process η(0)(t):
T: (a) p
(ε)
ij → p
(0)
ij as ε → 0, i �= 0, j ∈ X;
(b) F
(ε)
ij (·) ⇒ F
(0)
ij (·) as ε → 0, i �= 0, j ∈ X.
Here and henceforth, the symbol ⇒ is used to denote weak convergence
for distribution functions.
Let us introduce the hitting-without-absorption probabilities 0f
(ε)
ij =
Pi{ν(ε)
j < ν
(ε)
0 }, i, j �= 0.
We assume that the set X0 = {j �= 0} is a class of recurrent-without-
absorption states for the limiting semi-Markov process, i.e., the following
condition holds:
E: 0f
(0)
ij > 0 for all i, j �= 0.
Let us introduce distribution functions
0G
(ε)
ij (t) = Pi{μ(ε)
0 ≤ t, ν
(ε)
j < ν
(ε)
0 }, t ≥ 0,
and the following mixed power-exponential moment generating functions,
for i, j �= 0, n = 0, 1, . . .,
ψ
(ε)
ij [ρ, n] =
∫ ∞
0
tneρtF
(ε)
ij (dt), ρ ≥ 0,
and
0φ
(ε)
ij [ρ, n] =
∫ ∞
0
tneρt
0G
(ε)
ij (dt) = Ei(μ
(ε)
0 )neρμ
(ε)
0 χ(ν
(ε)
j < ν
(ε)
0 ), ρ ≥ 0.
We assume also the following conditions:
C1: There exists δ > 0 such that lim0≤ε→0ψ
(ε)
ij [δ, 0] < ∞, i �= 0, j ∈ X,
and
C2: There exist i �= 0 and βi ∈ (0, δ], for δ defined in condition C1, such
that 0φ
(0)
ii [βi, 0] ∈ (1,∞).
Let us consider the following characteristic equations, for i �= 0,
0φ
(0)
ii [ρ, 0] = 1. (2)
270 DMITRII S. SILVESTROV
It can be shown that conditions I, A, T, E, and C1, C2 imply that there
exists ε1 > 0 such that for every ε ≤ ε1: (a) equation (2) have a unique
root ρ(ε) ≥ 0; (b) the root ρ(ε) does not depend of the choice of the state
i �= 0; (c) there exists β < δ such that ρ(ε) < β; (d) ρ(ε) → ρ(0) as ε → 0;
(e) ρ(0) > 0 if and only if
∑
i�=0 p
(0)
i0 > 0.
Let also assume the following non-arithmetic condition:
N: There exists i �= 0 such that the distribution function 0G
(ε)
ii (t) is non-
arithmetic.
It can be shown that conditions I, A, T, and E imply that 0G
(ε)
ii (t) is
arithmetic or non-arithmetic distribution function for all i �= 0 simultane-
ously.
It can be shown that conditions I, A, T, E, N, and C1, C2 imply that
there exist ε1 ≥ ε2 > 0 such that for every ε ≤ ε2 the following limits exist
for i �= 0,
lim
t→∞
Pi{η(ε)(t) = l/μ
(ε)
0 > t} = π
(ε)
l (ρ(ε)), l �= 0. (3)
The limits satisfy the following conditions: (f) they do not depend on
i �= 0; (g) π
(ε)
l (ρ(ε)) > 0, l �= 0; (h)
∑
l �=0 π
(ε)
l (ρ(ε)) = 1. Relations (f) –
(h) clarify why it is natural to call the distribution π
(ε)
l (ρ(ε)), l �= 0 a quasi-
stationary distribution for the semi-Markov process η(ε)(t) with absorption
times μ
(ε)
0 .
Also let us consider the following mixed power-exponential moment gen-
erating functions, for i, j, l �= 0 and n = 0, 1, . . ..
ω
(ε)
ijl [ρ, n] =
∫ ∞
0
sneρsPi{η(ε)(s) = l, μ
(ε)
j ∧ μ
(ε)
0 > s}ds, ρ ≥ 0,
and
ω
(ε)
ij [ρ, n] =
∫ ∞
0
sneρsPi{μ(ε)
j ∧ μ
(ε)
0 > s}ds =
∑
l �=0
ω
(ε)
ijl [ρ, n], ρ ≥ 0.
It can be shown that, under conditions I, A, T, E and C1, C2 there
exists ε2 ≥ ε3 > 0 such that (i) ω
(ε)
ij [ρ, n] < ∞ for ρ ≤ β, i, j �= 0, n =
0, 1, . . . and ε ≤ ε3, and (j) the following formula takes place for the quasi-
stationary distribution, for every ε ≤ ε3 and state j �= 0,
π
(ε)
l (ρ(ε)) =
ω
(ε)
jjl[ρ
(ε), 0]
ω
(ε)
jj [ρ(ε), 0]
, l �= 0. (4)
Note that this formula defines the quasi-distribution even if condition
N does not hold.
Finally, let us introduce the following nonlinear perturbation conditions:
NONLINEARLY PERTURBED SEMI-MARKOV PROCESSES 271
P: p
(ε)
ij = p
(0)
ij + εeij [1] + · · ·+ εkeij [k] + o(εk) for i, j �= 0, where |eij[r]| <
∞, r = 1, . . . , k, i, j �= 0.
and, for ρ < β,
P[ρ]: ψ
(ε)
ij [ρ, n] = ψ
(0)
ij [ρ, n] + εvij[ρ, 1, n] + · · · + εk+1−nvij [ρ, k + 1 − n, n] +
o(εk+1−n) for n = 0, . . . , k + 1, i �= 0, j ∈ X, where |vij [ρ, r, n]| <
∞, r = 1, . . . , k + 1 − n, n = 0, . . . , k + 1, i �= 0, j ∈ X.
The following theorem presents the asymptotic expansions for quasi-
stationary distributions for nonlinearly perturbed semi-Markov processes.
Theorem 1. Let conditions I, A, T, E, C1, C2, P, and P[ρ(0)] hold. Then,
the quasi-stationary probabilities π
(ε)
l (ρ(ε)), l �= 0, which do not depend on
the choice of the state j �= 0 in formula (4), have the following asymptotic
expansion, for every l, j �= 0:
π
(ε)
l (ρ(ε)) =
ω
(0)
jjl [ρ
(0), 0] + εfjjl[ρ
(0), 1] + · · ·+ εkfjjl[ρ
(0), k] + o(εk)
ω
(0)
jj [ρ(0), 0] + εfjj[ρ(0), 1] + · · ·+ εkfjj[ρ(0), k] + o(εk)
(5)
= π
(0)
l (ρ(0)) + εgl[ρ
(0), 1] + · · · + εkgl[ρ
(0), k] + o(εk),
where the coefficients fjjl[ρ
(0), r], fjj[ρ
(0), r], gl[ρ
(0), r], r = 1, . . . , k are given
by explicit recurrence formulas as functions of coefficients in the expansions
penetrating the perturbation conditions P and P[ρ(0)].
3. Conclusion
In conclusion, I would like to note that this paper presents the result
from a new book written in cooperation with Professor Mats Gyllenberg.
The algorithm for evaluation of the coefficients in the asymptotic expansions
(5) and the proof of Theorem 1 is given in this book.
The book mentioned above is devoted to studies of quasi-stationary phe-
nomena in nonlinearly perturbed stochastic systems. The methods based
on exponential asymptotics for nonlinearly perturbed renewal equation are
used. Mixed ergodic and large deviation theorems are presented for nonlin-
early perturbed regenerative processes, semi-Markov processes and Markov
chains. Applications to nonlinearly perturbed population dynamics and
epidemic models, queueing systems and risk processes are considered. The
book also includes an extended bibliography of works in the area.
Bibliography
1. Gyllenberg, M and Silvestrov, D.S. Quasi-stationary Phenomena in Non-
linearly Perturbed Stochastic Systems (submitted).
Department of Mathematics and Physics,
Mälardalen University,
Box 883, SE-721 23 Väster̊as, Sweden.
E-mail address: dmitrii.silvestrov@mdh.se
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