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

We study the additive functional $X_n(\alpha)$ on conditioned Galton-Watson trees given, for arbitrary complex $\alpha$, by summing the $\alpha$th power of all subtree sizes. Allowing complex $\alpha$ is advantageous, even for the study of real $\alpha$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Re\alpha < 0$, we prove that $X_n(\alpha)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $\alpha$, 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 $\alpha$ in various regions of the complex plane. We focus mainly on the case where $\Re\alpha > 0$, for which $X_n(\alpha)$, suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution $\xi$ of the conditioned Galton-Watson tree, assuming only that $E[\xi] = 1$ and $0 < \mathrm{Var} [\xi] < \infty$. Under a weak extra moment assumption on $\xi$, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when $\Re\alpha > \frac12$, the limit random variable $Y(\alpha)$ can be expressed as a function of a normalized Brownian excursion.