# The first exit of almost strongly recurrent semi-Markov processes

Joachim Domsta; Franciszek Grabski

Applicationes Mathematicae (1995)

- Volume: 23, Issue: 3, page 285-304
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topDomsta, Joachim, and Grabski, Franciszek. "The first exit of almost strongly recurrent semi-Markov processes." Applicationes Mathematicae 23.3 (1995): 285-304. <http://eudml.org/doc/219132>.

@article{Domsta1995,

abstract = {Let $(·)$, n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding renewal kernels $$ on K×K to J×J are assumed to be suitably convergent to a renewal kernel P (on J×J). The HSMP on J corresponding to P is assumed to be strongly recurrent. Let [$π_j$; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged p.d.f. $F_\{ϑ\}(t) :=\sum _\{j,k ∈ J\} π_jP_\{j,k\}(t)$, t ∈ i$ℝ_+$, and its Laplace-Stieltjes transform $\widetilde\{F\}_ϑ$, the above assumptions imply: The time $\stackrel\{n\}\{T\}_\{J\}$ of the first exit of $\stackrel\{n\}\{X\}(·)$ from J has a limit p.d. (up to some constant factors) iff 1 - $\widetilde\{F\}_ϑ$ is regularly varying at 0 with a positive degree, say α ∈ (0,1]. Then the transform of the limit p.d.f. equals $\widetilde\{G\}^\{(α)\}(s) = (1+s^\{α\})^\{-1\}$, Re s ≥ 0. This extends the results by V. S. Korolyuk and A. F. Turbin (1976) obtained for α = 1 under essentially stronger conditions.},

author = {Domsta, Joachim, Grabski, Franciszek},

journal = {Applicationes Mathematicae},

keywords = {limit distribution; Markov renewal; first exit; extended exponential p.d; semi-Markov; recurrent Markov processes; semi-Markov processes; Markov renewal processes; extended exponential probability distribution},

language = {eng},

number = {3},

pages = {285-304},

title = {The first exit of almost strongly recurrent semi-Markov processes},

url = {http://eudml.org/doc/219132},

volume = {23},

year = {1995},

}

TY - JOUR

AU - Domsta, Joachim

AU - Grabski, Franciszek

TI - The first exit of almost strongly recurrent semi-Markov processes

JO - Applicationes Mathematicae

PY - 1995

VL - 23

IS - 3

SP - 285

EP - 304

AB - Let $(·)$, n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding renewal kernels $$ on K×K to J×J are assumed to be suitably convergent to a renewal kernel P (on J×J). The HSMP on J corresponding to P is assumed to be strongly recurrent. Let [$π_j$; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged p.d.f. $F_{ϑ}(t) :=\sum _{j,k ∈ J} π_jP_{j,k}(t)$, t ∈ i$ℝ_+$, and its Laplace-Stieltjes transform $\widetilde{F}_ϑ$, the above assumptions imply: The time $\stackrel{n}{T}_{J}$ of the first exit of $\stackrel{n}{X}(·)$ from J has a limit p.d. (up to some constant factors) iff 1 - $\widetilde{F}_ϑ$ is regularly varying at 0 with a positive degree, say α ∈ (0,1]. Then the transform of the limit p.d.f. equals $\widetilde{G}^{(α)}(s) = (1+s^{α})^{-1}$, Re s ≥ 0. This extends the results by V. S. Korolyuk and A. F. Turbin (1976) obtained for α = 1 under essentially stronger conditions.

LA - eng

KW - limit distribution; Markov renewal; first exit; extended exponential p.d; semi-Markov; recurrent Markov processes; semi-Markov processes; Markov renewal processes; extended exponential probability distribution

UR - http://eudml.org/doc/219132

ER -

## References

top- [1] S. Asmussen, Applied Probability and Queues, Wiley, Chichester, 1987. Zbl0624.60098
- [2] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971. Zbl0219.60003
- [3] I. B. Gertsbakh, Asymptotic methods in reliability theory : a review, Adv. Appl. Probab. 16 (1984), 147-175. Zbl0528.60085
- [4] J. Keilson, A limit theorem for passage times in ergodic regenerative processes, Ann. Math. Statist. 37 (1966), 866-870. Zbl0143.19101
- [5] B. Kopociński, An Outline of Renewal and Reliability Theory, PWN, Warszawa, 1973 (in Polish). Zbl0365.60003
- [6] V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and Their Applications, Naukova Dumka, Kiev, 1976 (in Russian). Zbl0371.60106
- [7] E. Seneta, Regularly Varying Functions, Lecture Notes in Math. 508, Springer, Berlin, 1976. Zbl0324.26002
- [8] D. S. Silvestrov, Semi-Markov Processes with a Discrete State Space, Sovetskoe Radio, Moscow, 1980 (in Russian).
- [9] A. D. Solovyev, Asymptotic behavior of the time of the first occurrence of a rare event, Engnrg. Cybernetics 9 (1971), 1038-1048.
- [10] A. D. Solovyev, Analytical Methods of the Reliability Theory, WNT, Warszawa, 1979 (in Polish).
- [11] A. F. Turbin, Applications of the inversion of linear operators perturbed on the spectrum to some asymptotic problems connected with Markov chains and semi-Markov processes, Teor. Veroyatnost. i Mat. Statist., Kiev, 1972 (in Russian). Zbl0339.47011

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.