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Level-crossing ordering of semi-Markov processes and markov chains

Part of: Special processes Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Fátima Ferreira  and
António Pacheco
Fátima Ferreira*
Affiliation:
CEMAT and Universidade de Trás os Montes e Alto Douro
António Pacheco*
Affiliation:
CEMAT, CLC, and Instituto Superior Técnico, UTL
*
∗Postal address: UTAD, Departamento de Matemática, Quinta dos Prados, Apartado 1013, 5001-911 Vila Real, Portugal. Email address: mmferrei@utad.pt
∗∗Postal address: Instituto Superior Técnico, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Email address: apacheco@math.ist.utl.pt
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Abstract

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We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

Keywords

Integral stochastic orderlevel-crossing orderingMarkov chainsample pathsemi-Markov processsimulationstochastic ordering

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60K15: Markov renewal processes, semi-Markov processes 68U20: Simulation
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 989 - 1002
Copyright
© Applied Probability Trust 2005 

References

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