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

Abstract

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