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IV: 12, 133-150, LNM 124 (1970)
MEYER, Paul-André
Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)
The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance Z. für W-theorie, 2, 1964), under the very restrictive assumption that the ``event'' has a non-zero probability to occur at fixed times. The general theory is due to Krylov and Yushkevich (Trans. Moscow Math. Soc., 13, 1965), a deep paper difficult to read and to apply in concrete cases. Hoffmann-Jørgensen (Math. Scand., 24, 1969) developed the theory under simple and efficient axioms. It is shown that a regenerative set defined axiomatically is the same thing as the set of returns of a strong Markov process to a fixed state, or the range of a subordinator
Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (Th. Prob. Appl., 16, 1971) and Maisonneuve, Systèmes Régénératifs, Astérisque 15, 1974
Keywords: Renewal theory, Regenerative sets, Recurrent events
Nature: Exposition
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