Einblick - Stochastic Processes - # Weak Convergence of Continuous-State Branching Processes with Large Immigration

Functional Limit Theorems for Continuous-State Branching Processes with Heavy-Tailed Immigration

Q: How do the results extend to the case where the branching mechanism has infinite mean (i.e., Ψ'(0+) = -∞)?

In the context of continuous-state branching processes with immigration (CBIs), the case where the branching mechanism has an infinite mean, indicated by Ψ'(0+) = -∞, presents significant challenges. The results established in the paper primarily focus on scenarios where the branching dynamics have a finite mean, allowing for a structured analysis of the immigration effects on the population dynamics. When Ψ'(0+) = -∞, the growth of the CBI process becomes almost surely super-exponential, leading to a scenario where the immigration dynamics may not be able to counterbalance the rapid growth of the branching process. In this regime, the immigration can become negligible compared to the explosive growth of the branching process, which complicates the identification of limiting behaviors. The authors suggest that the functional limit theorems for CBIs with infinite mean branching dynamics are left for future research. This indicates that while the current framework provides a robust foundation for analyzing cases with finite mean, the infinite mean scenario requires a different approach, potentially involving the study of extremal behaviors and the development of new mathematical tools to capture the dynamics of such processes.

Q: Can the authors' approach be adapted to study functional limit theorems for other types of stochastic processes with immigration, such as diffusion processes or Lévy processes?

Yes, the authors' approach can be adapted to study functional limit theorems for other types of stochastic processes with immigration, including diffusion processes and Lévy processes. The methodology employed in the paper, which involves the convergence of generators and the use of Poisson shot noise structures, is versatile and can be applied to a broader class of stochastic models. For diffusion processes, one could analyze the impact of immigration on the diffusion dynamics by considering the generator of the diffusion process and incorporating immigration terms analogous to those in the CBI framework. Similarly, for Lévy processes, the authors' techniques could be utilized to explore the interplay between branching and immigration by examining the Lévy-Khintchine representation and the associated immigration measures. The key would be to establish the appropriate conditions under which the immigration dynamics influence the limiting behavior of these processes, similar to the regimes identified for CBIs. By leveraging the insights gained from the current study, researchers could extend the functional limit theorems to encompass a wider array of stochastic processes, enriching the understanding of immigration effects across different modeling contexts.

Q: What are the potential applications of the identified limiting processes (subordinators, CBIs, extremal processes, extremal shot noise processes) in modeling real-world phenomena involving branching and immigration dynamics?

The identified limiting processes—subordinators, continuous-state branching processes (CBIs), extremal processes, and extremal shot noise processes—have numerous potential applications in modeling real-world phenomena that involve branching and immigration dynamics. Subordinators: These processes are often used to model phenomena with random waiting times, such as the time until the next event in a Poisson process. They can be applied in fields like finance for modeling stock prices or in biology for modeling the growth of populations with random immigration. Continuous-State Branching Processes (CBIs): CBIs are particularly relevant in biological contexts, such as modeling the growth of populations where individuals reproduce and new individuals immigrate into the population. They can also be applied in epidemiology to model the spread of diseases, where the branching represents the spread of infections and immigration accounts for new cases entering the population. Extremal Processes: These processes are useful in fields such as environmental science, where they can model the maximum levels of pollutants or extreme weather events over time. They can also be applied in finance to assess the risk of extreme losses in investment portfolios. Extremal Shot Noise Processes: These processes can model systems where the impact of rare but significant events (such as large immigration events) dominates the dynamics. Applications include telecommunications, where the arrival of data packets can be modeled as a shot noise process, and in risk management, where extreme losses need to be assessed. Overall, the insights gained from the study of these limiting processes can inform decision-making in various fields, including ecology, finance, and risk assessment, by providing a mathematical framework to understand the complex interactions between branching and immigration dynamics.

Kernkonzepte

Functional limit theorems are established for continuous-state branching processes with immigration, where the reproduction laws have finite first moments and the immigration laws exhibit heavy tails. Different regimes of immigration are identified, leading to limiting processes that are either subordinators, continuous-state branching processes with immigration, extremal processes, or extremal shot noise processes.

Zusammenfassung

The article establishes functional limit theorems for continuous-state branching processes with immigration (CBIs), where the reproduction laws have finite first moments and the immigration laws exhibit large tails.

The key highlights and insights are:

Two regimes of immigration are identified based on whether the immigration measure has a log moment or not.
When the immigration measure has a log moment, the limiting processes are either subordinators or CBIs, depending on the relationship between the branching and immigration mechanisms.
When the immigration measure has no log moment, the limiting processes are either extremal processes or extremal shot noise processes, depending on the precise asymptotic behavior of the immigration mechanism.
The proof techniques involve the convergence of generators, which allows the authors to handle different regimes in a unified manner.
The results generalize and complement previous work on functional limit theorems for Galton-Watson processes with immigration, establishing the analogous results in the continuous-state setting.
The article provides a comprehensive understanding of how the interplay between the branching and immigration dynamics can lead to various types of limiting processes in the context of continuous-state branching processes.

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Weak convergence of continuous-state branching processes with large immigration

by Clém... um arxiv.org 10-01-2024

https://arxiv.org/pdf/2311.04045.pdf

Weak convergence of continuous-state branching processes with large immigration

Tiefere Fragen

How do the results extend to the case where the branching mechanism has infinite mean (i.e., Ψ'(0+) = -∞)?

In the context of continuous-state branching processes with immigration (CBIs), the case where the branching mechanism has an infinite mean, indicated by Ψ'(0+) = -∞, presents significant challenges. The results established in the paper primarily focus on scenarios where the branching dynamics have a finite mean, allowing for a structured analysis of the immigration effects on the population dynamics. When Ψ'(0+) = -∞, the growth of the CBI process becomes almost surely super-exponential, leading to a scenario where the immigration dynamics may not be able to counterbalance the rapid growth of the branching process.
In this regime, the immigration can become negligible compared to the explosive growth of the branching process, which complicates the identification of limiting behaviors. The authors suggest that the functional limit theorems for CBIs with infinite mean branching dynamics are left for future research. This indicates that while the current framework provides a robust foundation for analyzing cases with finite mean, the infinite mean scenario requires a different approach, potentially involving the study of extremal behaviors and the development of new mathematical tools to capture the dynamics of such processes.

Can the authors' approach be adapted to study functional limit theorems for other types of stochastic processes with immigration, such as diffusion processes or Lévy processes?

Yes, the authors' approach can be adapted to study functional limit theorems for other types of stochastic processes with immigration, including diffusion processes and Lévy processes. The methodology employed in the paper, which involves the convergence of generators and the use of Poisson shot noise structures, is versatile and can be applied to a broader class of stochastic models.
For diffusion processes, one could analyze the impact of immigration on the diffusion dynamics by considering the generator of the diffusion process and incorporating immigration terms analogous to those in the CBI framework. Similarly, for Lévy processes, the authors' techniques could be utilized to explore the interplay between branching and immigration by examining the Lévy-Khintchine representation and the associated immigration measures.
The key would be to establish the appropriate conditions under which the immigration dynamics influence the limiting behavior of these processes, similar to the regimes identified for CBIs. By leveraging the insights gained from the current study, researchers could extend the functional limit theorems to encompass a wider array of stochastic processes, enriching the understanding of immigration effects across different modeling contexts.

What are the potential applications of the identified limiting processes (subordinators, CBIs, extremal processes, extremal shot noise processes) in modeling real-world phenomena involving branching and immigration dynamics?

The identified limiting processes—subordinators, continuous-state branching processes (CBIs), extremal processes, and extremal shot noise processes—have numerous potential applications in modeling real-world phenomena that involve branching and immigration dynamics.

Subordinators: These processes are often used to model phenomena with random waiting times, such as the time until the next event in a Poisson process. They can be applied in fields like finance for modeling stock prices or in biology for modeling the growth of populations with random immigration.

Continuous-State Branching Processes (CBIs): CBIs are particularly relevant in biological contexts, such as modeling the growth of populations where individuals reproduce and new individuals immigrate into the population. They can also be applied in epidemiology to model the spread of diseases, where the branching represents the spread of infections and immigration accounts for new cases entering the population.

Extremal Processes: These processes are useful in fields such as environmental science, where they can model the maximum levels of pollutants or extreme weather events over time. They can also be applied in finance to assess the risk of extreme losses in investment portfolios.

Extremal Shot Noise Processes: These processes can model systems where the impact of rare but significant events (such as large immigration events) dominates the dynamics. Applications include telecommunications, where the arrival of data packets can be modeled as a shot noise process, and in risk management, where extreme losses need to be assessed.

Overall, the insights gained from the study of these limiting processes can inform decision-making in various fields, including ecology, finance, and risk assessment, by providing a mathematical framework to understand the complex interactions between branching and immigration dynamics.