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On the Superposition and Thinning of Generalized Counting Processes

M. Dhillon, K. K. Kataria

TL;DR

This work analyzes the superposition (merging) and thinning (splitting) of generalized counting processes (GCPs). It proves that the sum of independent GCPs is itself a GCP with jump sizes up to the maximum component and arrival rates given by appropriate sums of the components’ rates; convergence for countably many GCPs requires finiteness of these sums. It further derives jump-origin probabilities, shows that jump packets in the merged process obey a Poisson process, and develops two splitting schemes: Type I (independent splits with reduced rates) and Type II (splits allowing simultaneous jumps and generally inducing dependence). The results are demonstrated through applications to industrial fishing and hotel booking management, illustrating how merged and split GCPs yield tractable total and type-specific counts with closed-form moments and conditioning properties.

Abstract

In this paper, we study the merging and splitting of generalized counting processes (GCPs). First, we study the merging of a finite number of independent GCPs and then extend it to the case of countably infinite. The merged process is observed to be a GCP with increased arrival rates. It is shown that a packet of jumps arrives in the merged process according to the Poisson process. Also, we study two different types of splitting of a GCP. In the first type, we study the splitting of jumps of a GCP where the probability of simultaneous jumps in the split components is negligible. In the second type, we consider the splitting of jumps in which there is a possibility of simultaneous jumps in the split components. It is shown that the split components are GCPs with certain decreased jump rates. Moreover, the independence of split components is established. Later, we discuss applications of the obtained results to industrial fishing problem and hotel booking management system.

On the Superposition and Thinning of Generalized Counting Processes

TL;DR

This work analyzes the superposition (merging) and thinning (splitting) of generalized counting processes (GCPs). It proves that the sum of independent GCPs is itself a GCP with jump sizes up to the maximum component and arrival rates given by appropriate sums of the components’ rates; convergence for countably many GCPs requires finiteness of these sums. It further derives jump-origin probabilities, shows that jump packets in the merged process obey a Poisson process, and develops two splitting schemes: Type I (independent splits with reduced rates) and Type II (splits allowing simultaneous jumps and generally inducing dependence). The results are demonstrated through applications to industrial fishing and hotel booking management, illustrating how merged and split GCPs yield tractable total and type-specific counts with closed-form moments and conditioning properties.

Abstract

In this paper, we study the merging and splitting of generalized counting processes (GCPs). First, we study the merging of a finite number of independent GCPs and then extend it to the case of countably infinite. The merged process is observed to be a GCP with increased arrival rates. It is shown that a packet of jumps arrives in the merged process according to the Poisson process. Also, we study two different types of splitting of a GCP. In the first type, we study the splitting of jumps of a GCP where the probability of simultaneous jumps in the split components is negligible. In the second type, we consider the splitting of jumps in which there is a possibility of simultaneous jumps in the split components. It is shown that the split components are GCPs with certain decreased jump rates. Moreover, the independence of split components is established. Later, we discuss applications of the obtained results to industrial fishing problem and hotel booking management system.
Paper Structure (20 sections, 13 theorems, 106 equations, 3 figures)

This paper contains 20 sections, 13 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Merging of independent GCPs
  3. Merging of finitely many independent GCPs
  4. Merging of countably many independent GCPs
  5. Jump probabilities from a particular process
  6. The case of merging of finitely many GCPs
  7. The case of merging of countably many GCPs
  8. Jumps packet from a particular process
  9. Splitting of jumps in GCP
  10. Splitting of jumps in GCP- Type I
  11. Splitting of jumps with a coin flip
  12. Splitting of jumps on rolling a $q$-faced die
  13. Splitting of jumps in GCP- Type II
  14. Splitting of jumps in two component processes
  15. Splitting jumps on rolling a $q$-faced die
  16. ...and 5 more sections

Key Result

Proposition 2.1

The merged process $\{\mathscr{M}^2(t)\}_{t\geq0}$ has independent and stationary increments.

Figures (3)

  • Figure 1: Joint probabilities of the merging components in an infinitesimal interval of length $h$.
  • Figure 2: Merging of $q$ independent GCPs.
  • Figure 3: Splitting of GCP with $k=5$.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 18 more