Galton-Watson processes, simple varieties of trees and Khinchin families

TL;DR

This paper introduces a unified framework connecting trees, Galton-Watson processes, and Khinchin families, providing new formulas for extinction probabilities and a method for efficient simulation.

Contribution

It offers a novel analytic approach using Lagrange's inversion to unify combinatorial and probabilistic models, simplifying classical methods and enabling efficient simulations.

Findings

Derived new coefficient-based expressions for extinction probabilities.

Reinterpreted boundary phenomena in terms of inverse solutions to Lagrange's equation.

Developed a computationally efficient simulation method for Galton-Watson processes.

Abstract

In this note, we introduce a unified analytic framework that connects simple varieties of trees, Bienayme-Galton-Watson processes and Khinchin families. Using Lagrange's inversion formula, we derive new coefficient-based expressions for extinction probabilities and reinterpret them as boundary phenomena tied to the domain of the inverse of the solution to Lagrange's equation. This perspective reveals an additional link between combinatorial and probabilistic models, simplifying classical arguments and yielding new results. It also leads to a computationally efficient method for simulating Galton-Watson processes via power series coefficients.