Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

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Abstract

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Lévy subordinator and the inverse of the Lévy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

Keywords

Lévy subordinator Fractional Poisson process Simulation 

Mathematics Subject Classification (2010)

60G22 60G55 
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Notes

Acknowledgements

A part of this work was done while the second author was visiting the Department of Statistics and Probability, Michigan State University, during Summer 2016. The authors thank the referee for some useful comments.

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Authors and Affiliations

  1. 1.Operations Management and Quantitative Techniques AreaIndian Institute of Management IndoreIndoreIndia
  2. 2.Department of MathematicsIndian Institute of Technology BombayPowaiIndia

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