Abstract
A regenerative set M is a random set of real numbers that has a type of Markov property, not necessarily at arbitrary times like the set of instances following independent exponentially distributed waiting times, but only at those times belonging to M. Early uses of the term ‘regenerative set’ appear in [Meyer 1970] and [Maisonneuve and Morando 1970] — and, earlier, Tackás [1956] used ‘regeneration point’ for a member of a certain type of regenerative set. As clarification for some readers but possibly prematurely for others, I mention that Feller [1968], in Chapter XIII, used the term ‘recurrent event’, but this term has been largely discarded because a regenerative set is not what is usually called an event. Hoffman-Jørgensen [1969] uses the term ‘Markov set’, whereas Krylov and Yushkevich [1965] use ‘Markov random set’ for a stochastic process related to a regenerative set. Breiman [1968] and Kingman [1972] focus on the indicator function of M, using the terms ‘renewal process’ and ‘regenerative phenomenon’, respectively. We will stick to ‘regenerative set’.
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Fristedt, B. (1996). Intersections and Limits of Regenerative Sets. In: Aldous, D., Pemantle, R. (eds) Random Discrete Structures. The IMA Volumes in Mathematics and its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0719-1_9
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