Conditioned Galton–Watson Trees: The Shape Functional, and More on the Sum of Powers of Subtree Sizes and Its Mean

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2024年3月29日

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James Allen Fill,
Svante Janson,
Stephan Wagner

摘要

For a complex number $\alpha $ $α$ , we consider the sum of the $\alpha $ $α$ th powers of subtree sizes in Galton–Watson trees conditioned to be of size n. Limiting distributions of this functional $X_n(\alpha )$ $X_{n} (α)$ have been determined for ${\text {Re}}\alpha \ne 0$ $Re α \neq 0$ , revealing a transition between a complex normal limiting distribution for ${\text {Re}}\alpha < 0$ $Re α < 0$ and a non-normal limiting distribution for ${\text {Re}}\alpha > 0$ $Re α > 0$ . In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case ${\text {Re}}\alpha = 0$ $Re α = 0$ . The same results are also established in the case of the so-called shape functional $X_n'(0)$ $X_{n}^{'} (0)$ , which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. In addition, we prove convergence of all moments in the case ${\text {Re}}\alpha < 0$ $Re α < 0$ , where this result was previously missing, and establish new results about the asymptotic mean for real $\alpha < 1/2$ $α < 1 / 2$ .

A novel feature for ${\text {Re}}\alpha =0$ $Re α = 0$ is that we find joint convergence for several $\alpha $ $α$ to independent limits, in contrast to the cases ${\text {Re}}\alpha \ne 0$ $Re α \neq 0$ , where the limit is known to be a continuous function of $\alpha $ $α$ . Another difference from the case ${\text {Re}}\alpha \ne 0$ $Re α \neq 0$ is that there is a logarithmic factor in the asymptotic variance when ${\text {Re}}\alpha =0$ $Re α = 0$ ; this holds also for the shape functional.

The proofs are largely based on singularity analysis of generating functions.