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On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain

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Abstract

We propose a continuous-time adaptation of the well-known concept of success runs by considering a marked point process with two types of marks (success-failure) that appear according to an appropriate continuous-time Markov chain. By constructing a bivariate imbedded process (consisting of a run-counting and a phase process), we offer recursive formulas and generating functions for the distribution of the number of runs and the waiting time until the appearance of the n-th success run. We investigate the three most popular counting schemes: (i) overlap** runs of length k, (ii) non-overlap** runs of length k and (iii) runs of length at least k. We also present examples of applications regarding: the total penalty cost in a maintenance reliability system, the number of risky situations in a non-life insurance portfolio and the number of runs of increasing (or decreasing) asset price movements in high-frequency financial data.

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Authors and Affiliations

  1. Department of Statistics & Insurance Science, University of Piraeus, Piraeus, Greece

    Boutsikas V. Michael

  2. Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

    Vaggelatou Eutichia

Authors
  1. Boutsikas V. Michael
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  2. Vaggelatou Eutichia
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Correspondence to Boutsikas V. Michael.

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Michael, B.V., Eutichia, V. On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain. Methodol Comput Appl Probab 22, 969–993 (2020). https://doi.org/10.1007/s11009-019-09743-3

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  • DOI: https://doi.org/10.1007/s11009-019-09743-3

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