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Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems


  • Bretó, Carles
  • Ionides, Edward L.


We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.

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  • Bretó, Carles & Ionides, Edward L., 2011. "Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems," DES - Working Papers. Statistics and Econometrics. WS ws111914, Universidad Carlos III de Madrid. Departamento de Estadística.
  • Handle: RePEc:cte:wsrepe:ws111914

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    Cited by:

    1. Bretó, Carles, 2012. "Time changes that result in multiple points in continuous-time Markov counting processes," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2229-2234.
    2. Bretó, Carles, 2012. "On the infinitesimal dispersion of multivariate Markov counting systems," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 720-725.
    3. Bretó, Carles, 2014. "Trajectory composition of Poisson time changes and Markov counting systems," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 91-98.

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