Fractional Poisson Processes of Order k and Beyond

Neha Gupta () and Arun Kumar ()
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Neha Gupta: Indian Statistical Institute Delhi
Arun Kumar: Indian Institute of Technology Ropar

Journal of Theoretical Probability, 2023, vol. 36, issue 4, 2165-2191

Abstract: 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.

Keywords: Time-fractional Poisson process; Poisson process of order k; Space-fractional Poisson process; Infinite divisibility; Homogeneous Poisson field; Watanabe martingale characterization; 60G22; 60G51; 60G55 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-023-01268-3

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