Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution

TL;DR

The paper analyzes supercritical Galton–Watson processes in the near-critical limit $\lambda \downarrow 1$ and shows that the population size $Z_n$ scaled by $\lambda^n$ is well approximated by a compound Poisson–gamma (CPG) distribution. Using a perturbative approach to the cumulant generating function, the authors derive a universal leading-order form dependent only on the offspring variance $\kappa_2^{*}$ and map it to a CP.G. with parameters $\mu=\frac{2(\lambda-1)}{\kappa_2^{*}}$, $\alpha=1$, $\tau=\frac{\kappa_2^{*}}{2(\lambda-1)}$, valid under $Z_0=1$ (Condition I); they extend to random $Z_0$ (Condition II) where the limit is a CP.G. with updated mean $\mu=\frac{2\lambda_0(\lambda-1)}{\kappa_2^{*}}$. Numerical verifications using Poisson and geometric offspring distributions show good agreement in the bulk for large generations and under reasonable parameter regimes, with fits improving when $Z_0$ is random and $\lambda$ remains near 1. The work also discusses tail behaviors, diffusion-approximation links, and large-$n$ asymptotics, arguing that CP.G. provides a practically useful, analytically tractable model for cascaded multiplication processes such as electron multiplier signals. Overall, the CP.G. framework offers a principled, universal approximation for the distribution of population sizes in near-critical GW processes with wide applicability to physical and biological cascades.

Abstract

We study asymptotic properties of supercritical Galton-Watson (GW) branching processes in the asymptotic where the mean of the offspring distribution approaches 1 from above. We show that the population-size distribution of the GW branching processes at a sufficiently large generation in this asymptotic can be approximated by a compound Poisson-gamma distribution. Numerical experiments revealed that the compound Poisson-gamma models were in good agreement with the corresponding GW models for sufficiently large generations under a reasonable parameter regime. Our results can be regarded as supporting the use of the compound Poisson-gamma model as a model for cascaded multiplication processes, such as detection signals of electron multipliers and population sizes of individuals with specific biological characteristics.

Figures (5)

• Figure 1: Comparison between $P_{\mathrm{CPG}}$ and $P_n$ in the Poisson case under Condition I. Three different values of $\lambda$, $1.1,1.5,5.0$, are examined from top to bottom. The left column is for the plot of $P_n(0)$ (joined blue circles) against $n$, compared with $P_{\mathrm{CPG}}(0)$ (black line). The other columns compare $\lambda^n P_n(\lambda^n x)$ and $P_{\mathrm{CPG}}(x)$ in the semi-logarithmic scale. The bulk region and the right tail of the distribution are exhibited in the center and right columns, respectively.
• Figure 2: Comparison between $P_{\mathrm{CPG}}$ and $P_n$ in the geometric case under Condition I. Three different values of $\lambda$, $1.1,1.5,5.0$, are examined from top to bottom. The left column is for the plot of $P_n(0)$ (joined blue circles) against $n$, compared with $P_{\mathrm{CPG}}(0)$ (black line). The other columns compare $\lambda^n P_n(\lambda^n x)$ (joined color markers) and $P_{\mathrm{CPG}}(x)$ (black line) in the semi-logarithmic scale. The bulk region and the right tail of the distribution are exhibited in the center and right columns, respectively.
• Figure 3: Comparison between $P_{\mathrm{CPG}}$ and $Q_n$ in the Poisson case under Condition II at $\lambda=1.1$. Three different values of $\lambda_0$, $1,2,5$, are examined from top to bottom. The left columns are for the plot of $Q_n(0)$ (joined blue circles) against $n$, compared with $P_{\mathrm{CPG}}(0)$ (black line). The other columns compare $\lambda^n Q_n(\lambda^n x)$ and $P_{\mathrm{CPG}}(x)$ in the semi-logarithmic scale. The bulk region and the right tail of the distribution are exhibited in the center and right columns, respectively.
• Figure 4: Comparison between $P_{\mathrm{CPG}}$ and $Q_n$ in the Poisson case at $\lambda_0=1$. Three different values of $\lambda$, $1.5,2,5$, are examined from left to right. The top row gives the plot of $Q_n(0)$ (joined blue circles) against $n$, compared with $P_{\mathrm{CPG}}(0)$ (black line). The bottom row compare $\lambda^n Q_n(\ell=\lambda^n x)$ (joined color markers) and $P_{\mathrm{CPG}}(x)$ (black line) in the semi-logarithmic scale.
• Figure 5: Fitted $P_{\mathrm{CPG}}$ (black line), $Q_n$ for large $n$ (joined red squares), and $P_{\mathrm{CPG}}$ with the theoretical parameter values (blue dashed line) are compared in the Poisson case with $\lambda$ fairly larger than $unity$. Three different parameter sets, $(\lambda_0,\lambda)=(1,5)$, $(12,2.7)$, and $(100,2)$ from left to right, are considered. In all the cases, the fitted CPG distribution recovers a nice agreement with $Q_n$.

Theorems & Definitions (7)

• Proposition 1
• Proposition 2: Proposition 3 in Biggins1981
• proof
• Lemma 1
• proof
• Lemma 2
• proof