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

Abstract

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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 ) : = ∑ j , k ∈ J π j P j , k ( t ) , t ∈ i ℝ + , and its Laplace-Stieltjes transform F ˜ ϑ , the above assumptions imply: The time T n J of the first exit of X n ( · ) from J has a limit p.d. (up to some constant factors) iff 1 - 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 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.

How to cite

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Domsta, 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

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  10. [10] A. D. Solovyev, Analytical Methods of the Reliability Theory, WNT, Warszawa, 1979 (in Polish). 
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