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Interacting point processes

Fabrizio Cinque, Enzo Orsingher

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.

Interacting point processes

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.
Paper Structure (10 sections, 7 theorems, 88 equations)

This paper contains 10 sections, 7 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Interacting Skellam-type processes
  3. Generalizations
  4. Multiple interacting processes
  5. Bidimensional generalized Skellam-type interacting processes
  6. Interacting birth-death processes
  7. Death-migration process
  8. Difference of independent birth-death processes
  9. Time-changed interacting point processes
  10. Proof of equation \ref{['relazioneFormaAlternativaNumeroArmonico']}

Key Result

Theorem 2.1

Let $(N_1,N_2)$ be the stochastic vector defined in definizioneInfinitesimaleCoppiaTipoSkellam. Then, where the terms are independent non-homogeneous (classical) Skellam processes

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2: Covariance
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 14 more