2022/12/21 by James Allen Fill, Svante Janson, Fill, James Allen +3
Mathematics · Physics and Astronomy · #05C05 #60C05 #60F05 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2212.10871
openalex publication_date 2022/12/21 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
For a complex number α, we consider the sum of the αth powers of subtree sizes in Galton--Watson trees conditioned to be of size n. Limiting distributions of this functional Xn(α) have been determined for \Reα≠ 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 Xn'(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≠0, where the limit is known to be a continuous function of α. Another difference from the case \Reα≠0 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.