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Fractional Poisson Processes of Order k and Beyond

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Abstract

In this article, we introduce fractional Poisson fields of order k in n-dimensional Euclidean space of positive real valued vectors. We also work on time-fractional Poisson process of order k, space-fractional Poisson processes of order k and a tempered version of time-space fractional Poisson processes of order k. We discuss generalized fractional Poisson processes of order k in terms of Bernstein functions. These processes are defined in terms of fractional compound Poisson processes. The time-fractional Poisson process of order k naturally generalizes the Poisson process and the Poisson process of order k to a heavy-tailed waiting-times counting process. The space-fractional Poisson process of order k allows on average an infinite number of arrivals in any interval. We derive the marginal probabilities governing difference–differential equations of the introduced processes. We also provide the Watanabe martingale characterization for some time-changed Poisson processes.

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Acknowledgements

The authors are grateful to the reviewers for several helpful comments and suggestions, which have led to improvements in the paper. N.G. is supported by the institute research fellowship of Indian Statistical Institute Delhi, India. Further, A.K. would like to express his gratitude to Science and Engineering Research Board (SERB), India for financial support under the MATRICS research grant MTR/2019/000286.

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  1. Department of Theoretical Statistics and Mathematics Unit, Indian Statistical Institute Delhi, New Delhi, 110016, India

    Neha Gupta

  2. Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab, 140001, India

    Arun Kumar

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  1. Neha Gupta
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Gupta, N., Kumar, A. Fractional Poisson Processes of Order k and Beyond. J Theor Probab 36, 2165–2191 (2023). https://doi.org/10.1007/s10959-023-01268-3

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