Abstract
We obtain the explicit expressions for the state probabilities of various state-dependent versions of fractional point processes. The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference-differential equations governing the state probabilities of these state-dependent processes. The distributions of some convolutions of the Mittag-Leffler random variables are derived as special cases of the obtained results.
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Acknowledgements
A part of this work was done while the K. K. Kataria was visiting Institut des Hautes Études Scientifiques to attend a workshop, and P. Vellaisamy was visiting the Department of Statistics and Probability, Michigan State University, during summer 2017. The authors would like to thank the institutes for their hospitality and for providing a suitable research environment. Finally, the authors thank the referee for some helpful comments.
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Kataria, K.K., Vellaisamy, P. On Distributions of Certain State-Dependent Fractional Point Processes. J Theor Probab 32, 1554–1580 (2019). https://doi.org/10.1007/s10959-018-0835-z
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DOI: https://doi.org/10.1007/s10959-018-0835-z
Keywords
- State-dependent fractional pure birth process
- State-dependent time fractional Poisson process
- Time fractional Poisson process