The first exit of almost strongly recurrent semi-Markov processes

Domsta, Joachim; Grabski, Franciszek

Applicationes Mathematicae, Tome 23 (1995), p. 285-304 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $(\cdot)$ , 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. $F_{ϑ} (t) : = \sum_{j, k \in J} π_{j} P_{j, k} (t)$ , t ∈ i $ℝ_{+}$ , and its Laplace-Stieltjes transform ${\tilde{F}}_{ϑ}$ , the above assumptions imply: The time ${\overset{n}{T}}_{J}$ of the first exit of $\overset{n}{X} (\cdot)$ from J has a limit p.d. (up to some constant factors) iff 1 - ${\tilde{F}}_{ϑ}$ is regularly varying at 0 with a positive degree, say α ∈ (0,1]. Then the transform of the limit p.d.f. equals ${\tilde{G}}^{(α)} (s) = {(1 + s^{α})}^{- 1}$ , Re s ≥ 0. This extends the results by V. S. Korolyuk and A. F. Turbin (1976) obtained for α = 1 under essentially stronger conditions.

Publié le : 1995-01-01

Zbl 0836.60095

EUDML-ID : urn:eudml:doc:219132

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     author = {Joachim Domsta and Franciszek Grabski},
     title = {The first exit of almost strongly recurrent semi-Markov processes},
     journal = {Applicationes Mathematicae},
     volume = {23},
     year = {1995},
     pages = {285-304},
     zbl = {0836.60095},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv23i3p285bwm}
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Domsta, Joachim; Grabski, Franciszek. The first exit of almost strongly recurrent semi-Markov processes. Applicationes Mathematicae, Tome 23 (1995) pp. 285-304. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv23i3p285bwm/

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