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

TL;DR

This paper derives the right tail asymptotic series for densities of Galton-Watson processes with fractional generating functions, revealing fractal structures and convergence conditions.

Contribution

It introduces a novel asymptotic series for these densities and analyzes the fractal nature of exponential frequency structures in the complex plane.

Findings

Asymptotic series converges under specific conditions.

Fractal structures appear in the exponential factors.

Comparison with standard integral and left tail asymptotics shows new insights.

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.