The sum of powers of subtree sizes for conditioned Galton–Watson trees

Abstract

We study the additive functional $X_n(\mathit{\alpha })$ on conditioned Galton–Watson trees given, for arbitrary complex α, by summing the αth power of all subtree sizes. Allowing complex α is advantageous, even for the study of real α, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $Re\mathit{\alpha }<0$, we prove that $X_n(\mathit{\alpha })$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in α, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence extensions of, limiting distribution results for α in various regions of the complex plane. We focus mainly on the case where $Re\mathit{\alpha }>0$, for which $X_n(\mathit{\alpha })$, suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution ξ of the conditioned Galton–Watson tree, assuming only that $\mathbb{E}\mathrm{\xi }=1$ and $0<Var\mathrm{\xi }<\mathrm{\infty }$. Under a weak extra moment assumption on ξ, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when $Re\mathit{\alpha }>\frac{1}{2}$, the limit random variable $Y(\mathit{\alpha })$ can be expressed as a function of a normalized Brownian excursion.