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Ergodicity and inequalities in a class of point processes

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  • Lindvall, Torgny

Abstract

In this paper we extend some results from renewal theory to a certain class of point processes. Firstly, a strong version of Blackwell's renewal theorem is shown to hold when the memory of the process at time t, say, contains only the configuration of the latest m points of occurrence preceding t, and of the points in the interval [t-A,t], where A is a constant. Secondly, a generalization of the decreasing failure rate (DFR) concept is introduced, based on the following principle: "if there have been many points of occurrence recently, then we will soon experience another one". Inequalities and monotonicity results are established under this new type of DFR assumption.

Suggested Citation

  • Lindvall, Torgny, 1988. "Ergodicity and inequalities in a class of point processes," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 121-131, November.
    • Handle: RePEc:eee:spapps:v:30:y:1988:i:1:p:121-131
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    Cited by:

    1. Last, Günter & Szekli, Ryszard, 1999. "Time and Palm stationarity of repairable systems," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 17-43, January.
    2. Maxime Morariu-Patrichi & Mikko S. Pakkanen, 2017. "Hybrid marked point processes: characterisation, existence and uniqueness," Papers 1707.06970, arXiv.org, revised Oct 2018.
    3. Last, Günter, 1996. "Coupling with compensators," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 147-170, December.

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