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Asymptotic normality for general subtree counts in conditioned Galton--Watson trees

Fameno Rakotoniaina, Dimbinaina Ralaivaosaona

Abstract

Let $\mathcal{T}$ denote a Galton--Watson tree with offspring distribution $ξ$ satisfying $\mathbb{E}(ξ) = 1$, and let $\mathcal{T}_n$ be the Galton--Watson tree conditioned to have exactly $n$ nodes. We show that, under a mild moment condition on $ξ$, the number of occurrences of a fixed rooted plane tree $\mathbf{t}$ as a general subtree in $\mathcal{T}_n$ is asymptotically normal as $n \to \infty$, with both mean and variance linear in $n$. In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on $ξ$ is violated, the conclusion may fail.

Asymptotic normality for general subtree counts in conditioned Galton--Watson trees

Abstract

Let denote a Galton--Watson tree with offspring distribution satisfying , and let be the Galton--Watson tree conditioned to have exactly nodes. We show that, under a mild moment condition on , the number of occurrences of a fixed rooted plane tree as a general subtree in is asymptotically normal as , with both mean and variance linear in . In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on is violated, the conclusion may fail.
Paper Structure (5 sections, 10 theorems, 91 equations)

This paper contains 5 sections, 10 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the central limit theorem
  4. Non-degeneracy
  5. The moment assumption in Theorem \ref{['main theorem']}

Key Result

Theorem 1

Let $\mathcal{T}_n$ be the conditioned Galton--Watson tree with offspring distribution $\xi$, satisfying $\mathbb{E}(\xi^{2\Delta(\textbf{t})+1}) < \infty$. Then, as $n \to \infty$, the mean and variance of $N_{\textbf{t}}(\mathcal{T}_n)$ satisfy the asymptotic estimates where $\mu = \mathbb{E}(n_{\textbf{t}}(\mathcal{T}))$, and $\gamma \ge 0$ is a constant. Moreover, if then $\gamma \neq 0$, an

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 11 more