Conditioned Galton-Watson trees: the shape functional, and more on the sum of powers of subtree sizes and its mean (Q6564007)

This paper considers a conditioned Galton-Watson tree \(\mathcal{T}_n\) of size \(n\), and the random variables \[\NX_n(\alpha):=\sum_{v \in \mathcal{T}_n} |\mathcal{T}_{n, v}|^{\alpha},\N\]\Nwhere \(\mathcal{T}_{n, v}\) is the fringe subtree of \(\mathcal{T}_n\) rooted at a vertex \(v \in \mathcal{T}_n\) and the parameter \(\alpha\) is any complex number. Limiting distributions of this functional \(X_n(\alpha)\) have been determined for \(\operatorname{Re} \alpha \neq 0\), revealing a transition between a complex normal limiting distribution for \(\operatorname{Re} \alpha < 0\) and a non-normal limiting distribution for \(\operatorname{Re} \, \alpha > 0\). The results in this paper complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case \(\operatorname{Re} \alpha = 0\). In contrast to the case \(\operatorname{Re} \alpha \neq 0\) where the limit is known to be a continuous function of \(\alpha\), it is shown that \(X_n(\alpha)\) for different imaginary \(\alpha\) are asymptotically independent for \(\operatorname{Im} \alpha > 0\), and thus it is not possible to have joint convergence to a continuous random function. The parameter \(\alpha\) is allowed to be complex instead of just real as it allows the application of singularity analysis of generating functions. By keeping track of the leading terms in different regimes, interesting phenomena are identified. Of particular note is a logarithmic factor which appears asymptotically in the variance and in higher moments when \(\operatorname{Re} \alpha = 0\).\N\NWithin the case \(\operatorname{Re} \alpha = 0\), the subcase \(\alpha=0\) is trivial, since \(X_n(\alpha)=n\) is non-random. In this subcase, the authors instead study the shape functional \N\[\NX_n'(0)=\sum_{v \in \mathcal{T}_n} \log |\mathcal{T}_{n, v}|=\log \prod_{v \in \mathcal{T}_n} |\mathcal{T}_{n, v}|\N\]\Nand establish the same results. In particular, they fill the gap left by earlier work [the first and the second author, Electron. J. Probab. 27, Paper No. 114, 77 p. (2022; Zbl 1498.05057)] where moment convergence for \(X_n'(0)\) was only proved for \(\operatorname{Re} \alpha > 0\) and prove convergence of all moments in the case \(\operatorname{Re} \alpha < 0\). New results about the asymptotic mean for \(\operatorname{Re} \alpha < 1/2\) are also derived.\N\NThe paper ends with some intriguing open problems.