Branching Processes in Random Environments with Thresholds

Abstract

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.

Figures (3)

• Figure 1: In black weekly COVID cases in Italy from February 23, 2020 to February 3, 2023. In blue a BPRE starting with the same initial value and offspring mean having the negative binomial distribution with predefined number of successful trials $r=10$ and Gamma-distributed mean with shape parameter equal to the mean of the data and rate parameter $1$.
• Figure 2: In each column the numerical Experiments 1, 2, 3, and 4. In the first row, the process $Z_{n}$ for $n=10^4-10^2, \dots, 10^4$. In the second and third row, the empirical probability distribution of $\{ \Delta_{j}^{L} \}$ and $\{ \Delta_{j}^{U} \}$, respectively.
• Figure 3: In black the process $Z_{n}$ for $n \in \{0,1,\dots,10^{3} \}$, in red horizontal lines at $L_{0}$ and $L_{U}$, and in blue the thresholds $U_{j}$ and $L_{j}$. From left to right, the numerical Experiments 5, 6, 7, and 8.

Theorems & Definitions (47)

• Theorem 2.1
• Definition 1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.1
• proof : Proof of Theorem \ref{['theorem:extinction_probability']}
• proof : Proof of Theorem \ref{['theorem:recurrence_uniformly_ergodic_stationary_markov_chains']}