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 Xn(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 < Var xi < oo. 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 > 21, the limit random variable Y(alpha) can be expressed as a function of a normalized Brownian excursion