Limit theorems for critical branching processes in an extremely unfavorable random environment

Abstract

Let $\{Z_{m},m\geq 0\}$ be a critical branching process in random environment and $\{S_{m},m\geq 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an $α$-stable law we prove conditional limit theorems describing, as $n\rightarrow \infty $, the distribution the number of particles in the process $\{Z_{m},0\leq m\leq n\}$ given $Z_{n}>0$ and $S_{n}\leq const$.

Summary

This research paper delves into the behavior of critical branching processes within a random environment that consistently hinders growth. By conditioning on survival and a bounded associated random walk, the authors derive limit theorems characterizing the long-term distribution of particles, providing valuable insights into population dynamics under persistently adverse conditions.

Key Insights

• •The paper establishes the weak convergence of the conditional laws of the population size at time *n*, given survival and a bounded random walk, to a proper discrete probability distribution. This demonstrates that even in unfavorable environments, the population size converges to a predictable distribution.
• •A key finding is the weak convergence of a rescaled generation size process to a stochastic process with continuous paths under specific conditions. This offers a detailed characterization of the branching process's trajectory shape under these adverse conditions.
• •The authors build upon the theory of stable random walks and branching processes in random environments. They provide a rigorous mathematical framework involving renewal functions and delicate estimates to derive their limit theorems.

Practical Implications

• •The theoretical framework developed in this paper could be applied to modeling population dynamics in fluctuating environments, where conditions are frequently unfavorable for growth. This could aid in predicting population trends in scenarios like resource scarcity or environmental stress.
• •Future research could focus on relaxing the strong conditions imposed in the paper, such as B1 and B2, to broaden the applicability of the results to a wider range of branching processes in random environments.
• •Further investigation into the rate of convergence in the limit theorems and the sensitivity of the results to the parameter *K* would be valuable for practical applications, potentially leading to more accurate predictions and improved management strategies.

Links & Resources

Authors

Cite This Paper