[Paper Review] Critical Multi-Type Galton-Watson Trees Conditioned to be Large

Romain Abraham, Jean‐François Delmas+1 more|arXiv (Cornell University)|Nov 5, 2015

Stochastic processes and statistical mechanics16 references3 citations

TL;DR

This paper establishes the local convergence of critical multi-type Galton-Watson trees conditioned to have a large total progeny toward a multi-type Kesten’s tree under minimal conditions. By generalizing Neveu’s strong ratio limit theorem to d-dimensional aperiodic random walks and leveraging a multitype Dwass formula, the authors prove convergence when the mean matrix is primitive and the asymptotic type proportions match the normalized left Perron-Frobenius eigenvector.

ABSTRACT

Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .

Motivation & Objective

• To establish minimal conditions under which a critical multi-type Galton-Watson tree conditioned on large total progeny converges locally to a multi-type Kesten’s tree.
• To extend Neveu’s strong ratio limit theorem from dimension one to d-dimensional aperiodic random walks on ℤ^d.
• To generalize the Dwass formula to multi-type Galton-Watson forests using d-dimensional random walks.
• To provide a necessary and sufficient condition for local convergence in the multi-type setting under minimal moment assumptions.
• To address the open problem of conditioning on large total progeny without requiring exponential moments, especially in the absence of condensation phenomena.

Proposed method

• Generalize Neveu’s strong ratio limit theorem to d-dimensional aperiodic random walks on ℤ^d using uniform local limit theorems.
• Apply a multitype extension of the Dwass formula to encode critical multi-type Galton-Watson forests via d-dimensional random walks.
• Use the Legendre-Laplace transform and convex analysis to analyze the asymptotic behavior of the random walk increments.
• Establish uniform convergence of the local probabilities of the random walk using a d-dimensional local limit theorem.
• Prove the strong ratio limit property by bounding the relative likelihood of paths deviating from the mean drift.
• Use the convergence of the empirical measure of the offspring distribution to the Perron-Frobenius eigenvector to ensure correct asymptotic type proportions.

Experimental results

Research questions

• RQ1 Under what minimal conditions does a critical multi-type Galton-Watson tree conditioned on large total progeny converge locally to a multi-type Kesten’s tree?
• RQ2 Can the strong ratio limit theorem be extended from one-dimensional to d-dimensional random walks on ℤ^d for aperiodic distributions?
• RQ3 What is the role of the mean matrix’s Perron-Frobenius eigenvector in determining the asymptotic type proportions of the conditioned tree?
• RQ4 Is it possible to achieve local convergence without assuming exponential moments on the offspring distribution?
• RQ5 How does the generalized Dwass formula encode the joint distribution of multi-type tree sizes in terms of random walk paths?

Key findings

• The local limit of a critical multi-type Galton-Watson tree conditioned on large total progeny is the multi-type Kesten’s tree, provided the mean matrix is primitive and the asymptotic type proportions match the normalized left Perron-Frobenius eigenvector.
• The strong ratio limit theorem for d-dimensional aperiodic random walks is established, generalizing Neveu’s one-dimensional result.
• The proof relies on a uniform version of the d-dimensional local limit theorem for i.i.d. lattice-valued random vectors.
• The convergence holds under minimal assumptions: existence of the mean matrix and aperiodicity of the offspring distribution.
• The authors provide a new characterization of the Legendre-Laplace transform and its properties in a general convex analysis framework.
• The result extends previous work by Pénisson and Stephenson, which required higher moment or exponential moment conditions, by removing such assumptions under aperiodicity.