Limit theorems for sequences of random trees

TL;DR

This paper develops limit theorems for sequences of random trees using a new metric space framework, enabling statistical tests for comparing distributions of random trees.

Contribution

It introduces a metric-based approach to define mean trees and establishes laws of large numbers and central limit theorems in this setting.

Findings

Laws of large numbers for i.i.d. random trees

Central limit theorems for random trees

Proposed statistical tests for equality of tree distributions

Abstract

We consider a random tree and introduce a metric in the space of trees to define the ``mean tree'' as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.