Complete right tail asymptotic for the density of branching processes with fractional generating functions

TL;DR

The paper addresses the right-tail density $p(x)$ of the martingale limit for a supercritical Galton–Watson process with fractional generating functions, expressing $p(x)$ via a Fourier transform involving $\Pi$ that satisfies $G(\Pi(z))=\Pi(E z)$. It derives a complete right-tail asymptotic expansion $p(x)\sim -\sum_{\alpha}\operatorname{Res}(\Pi,\omega_{\alpha})\,e^{-\omega_{\alpha} x}$ as $x\to+\infty$, where the pole frequencies $\{\omega_{\alpha}\}$ arise from iterated preimages of zeros of $Q$ and form a fractal pattern in the complex plane; convergence is ensured under geometric-strip conditions $\vartheta>\pi$ and $\log_E r<-1$, with exact equality possible when $\deg P\le\deg Q+1$. The work compares this right-tail series with the complete left-tail asymptotics and the universal Fourier integral representation, clarifying when the asymptotic series can be an identity. Three rational generating function cases illustrate pole structures, residue calculations, and practical computation of $p(x)$ via $p_a(x)$ and $p_b(x)$, highlighting fractal spectral patterns in the pole distribution and demonstrating high-precision numerical agreement among representations.

Abstract

The right tail asymptotic series consisting of attenuating exponential terms are derived for the densities of Galton-Watson processes with fractional probability generating functions. The frequencies in the exponential factors form fractal structures in the complex plane. We discuss conditions when the asymptotic series converges everywhere. The obtained right tail asymptotic is compared with the standard integral representation of the density and with the complete left tail asymptotic.

Figures (9)

• Figure 1: The filled Julia set (black area) for $G(z)=0.9z+0.1z^6$ and its zoom at $z=1$. "Zoomed and shifted" Julia set divides ${\mathbb C}$ onto two sectors with different behavior of $\Pi(z)$.
• Figure 2: A fragment of the filled Julia set (black area) for $G$ defined in (\ref{['300']}).
• Figure 3: (a) The filled Julia set (black area) from Fig. \ref{['fig1']} near $z=1$ zoomed in about $1000$ times; (b) Distribution of poles of $\Pi(z)$, see (\ref{['002']}), in complex plane, for $G(z)$ defined in (\ref{['300']}).
• Figure 4: Comparison of three formulas (\ref{['002']}), (\ref{['012']}), and (\ref{['017']}) for the computation of the density $p$ in case of the probability generating function (\ref{['300']}).
• Figure 5: The filled Julia set for $G$ defined in (\ref{['400']}).