论文标题
在具有阈值的随机环境中的分支过程
Branching Processes in Random Environments with Thresholds
论文作者
论文摘要
通过应用于共同动力学的应用程序，我们描述了随机环境模型$ \ {z_n \} $中的分支过程，其特性在越过上和下阈值时会发生变化。这引入了一个周期性的路径行为，涉及增加和减少的时期，从而导致超临界和亚临界方案。即使该过程不是马可福音，我们在随机时间点上识别子序列$ \ {（τ_j，ν_j）\} $ - 特别是在交叉时间，{\ it {viz。}}，$ \ {（z__jj}，z___j}，z__j {viz。在轻度的力矩和规律性条件下，我们确定子序列具有再生结构，并证明在超临界和亚临界方面限制了该过程生长速率的正态分布。因此，我们建立了有关超批判性和亚临界政权的长度以及该过程在这些制度中花费的时间比例的限制定理。作为我们分析的副产品，我们从后代分布，阈值分布和环境序列的功能方面明确确定了限制差异。
Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(τ_j, ν_j)\}$ - specifically the values of the process at crossing times, {\it{viz.}}, $\{(Z_{τ_j}, Z_{ν_j})\}$ - along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.
