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Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching – Toward Application in Reliability

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Advances in Degradation Modeling

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

In this chapter, the weak convergence of additive functionals of processes with locally independent increments and with semi-Markov switching in the scheme of Poisson approximation is investigated. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness. This approach can be used in applications and especially in shock and degradation in random environment arising in reliability.

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Acknowledgements

The authors thank University of Bielefeld and personally Prof. Yu.G.Kondratiev for hospitality and financial support by DFG project 436 UKR 113/80/04-07.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Science, Kiev, Ukraine

    V.S. Koroliuk & I.V. Samoilenko

  2. Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, Compiègne, France

    N. Limnios

Authors
  1. V.S. Koroliuk
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  2. N. Limnios
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  3. I.V. Samoilenko
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Editor information

Editors and Affiliations

  1. Université Victor Segalen Bordeaux 2, I’lnstitut de Mathématiques de Bordeaux, Bordeaux Cedex, 33076, France

    M.S. Nikulin

  2. Département Génie Informatique, Labo. Mathématiques Appliquées Compiègne, Université de Technologie de Compiègne, Compiègne, France

    Nikolaos Limnios

  3. McMaster University, Department of Mathematics and Statistics, 1280 Main Street West, Hamilton, L8S 4K1, Canada

    N. Balakrishnan

  4. Fak. Mathematik, Inst. Mathematische Stochastik, Otto-von-Guericke-Universität, Universitätsplatz 2, Magdeburg, 39106, Germany

    Waltraud Kahle

  5. Laboratoire de Statistique Medicale, Université René Descartes, Paris 5, Paris, 75006, France

    Catherine Huber-Carol

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Koroliuk, V., Limnios, N., Samoilenko, I. (2010). Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching – Toward Application in Reliability. In: Nikulin, M., Limnios, N., Balakrishnan, N., Kahle, W., Huber-Carol, C. (eds) Advances in Degradation Modeling. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4924-1_7

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