Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton--Watson trees

TL;DR

This paper proves that certain additive functionals, including counts of specific fringe subtrees and protected nodes, are asymptotically normally distributed in conditioned Galton-Watson trees with finite variance offspring distribution.

Contribution

It establishes asymptotic normality for a broad class of additive functionals in conditioned Galton-Watson trees without requiring higher moment conditions.

Findings

Asymptotic normality of fringe subtree counts

Joint normality of multiple subtree counts

Normal distribution of protected nodes

Abstract

We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed.