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 $Realpha < 0$, 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 $Realpha > 0$, 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 $Realpha > frac12$, the limit random variable $Y(alpha)$ can be expressed as a function of a normalized Brownian excursion.