Interacting point processes

Fabrizio Cinque; Enzo Orsingher

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arXiv · 2510.12531

Interacting point processes

Fabrizio Cinque·Enzo Orsingher

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Abstract

We study two different types of vector point processes with interacting components, introducing a migration-type effect. The first case concerns two groups which modify their states with rate functions depending on time only. This yields a representation of the vector process in terms of independent non-homogeneous Skellam processes. In the general case, the decomposition involves independent Poisson processes. The second model is a birth-death-migration vector process. In the case of the linear death-migration we show that, for a fixed time instant, the vector is equal in distribution to the sum of two independent Multinomial random variables. As a byproduct we derive the distribution of a pure migration process. Finally, we study the described vector processes time-changed with the inverse of Bernstein subordinator, establishing a general result concerning the relatioship between fractional difference-differential equations and the probability mass function of a wider class of point processes.

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Fabrizio Cinque, Enzo Orsingher. 2025-10-14. Interacting point processes. https://arxiv.org/abs/2510.12531

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