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## Applicationes Mathematicae

1995-1996 | 23 | 3 | 285-304
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let $\stackrelnX(·)$, 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 $\stackrelnQ$ 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.
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EN
Czasopismo
Rocznik
Tom
Numer
Strony
285-304
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-08-05
poprawiono
1995-01-30
Twórcy
autor
• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk-Oliwa, Poland
autor
• Chair of Mathematics, Naval Academy, 81-919 Gdynia-Oksywie, Poland
Bibliografia
• [1] S. Asmussen, Applied Probability and Queues, Wiley, Chichester, 1987.
• [2] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971.
• [3] I. B. Gertsbakh, Asymptotic methods in reliability theory : a review, Adv. Appl. Probab. 16 (1984), 147-175.
• [4] J. Keilson, A limit theorem for passage times in ergodic regenerative processes, Ann. Math. Statist. 37 (1966), 866-870.
• [5] B. Kopociński, An Outline of Renewal and Reliability Theory, PWN, Warszawa, 1973 (in Polish).
• [6] V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and Their Applications, Naukova Dumka, Kiev, 1976 (in Russian).
• [7] E. Seneta, Regularly Varying Functions, Lecture Notes in Math. 508, Springer, Berlin, 1976.
• [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).
Typ dokumentu
Bibliografia
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