Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean

TL;DR

This paper completes the understanding of the distributional behavior of the sum of powers of subtree sizes in conditioned Galton-Watson trees, especially at the critical boundary where the real part of the power is zero, revealing new normal convergence and joint limits.

Contribution

It proves normal limiting distributions and moment convergence for the case 0 of the sum of powers of subtree sizes, and establishes new asymptotic mean results, extending previous work to the boundary case.

Findings

Normal limiting distribution for 0 case established.

Joint convergence to independent limits for multiple 0 powers.

Logarithmic factor in asymptotic variance when 0, unlike  cases.

Abstract

For a complex number $\alpha$, we consider the sum of the $\alpha$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(\alpha)$ have been determined for $\Re\alpha \neq 0$, revealing a transition between a complex normal limiting distribution for $\Re\alpha < 0$ and a non-normal limiting distribution for $\Re\alpha > 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Re\alpha = 0$. The same results are also established in the case of the so-called shape functional $X_n'(0)$, which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. Additionally, we prove convergence of all moments in the case $\Re\alpha < 0$, where this result was previously missing, and establish new results about the asymptotic mean for real $\alpha < 1/2$. A novel feature for $\Re\alpha=0$ is that we find joint convergence for several $\alpha$ to independent limits, in contrast to the cases $\Re\alpha\neq0$, where the limit is known to be a continuous function of $\alpha$. Another difference from the case $\Re\alpha\neq0$ is that there is a logarithmic factor in the asymptotic variance when $\Re\alpha=0$; this holds also for the shape functional. The proofs are largely based on singularity analysis of generating functions.