2021/04/06 by James Allen Fill, Svante Janson, Fill, James Allen +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #30E99 #60C05 #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Financial Risk and Volatility Modeling #Primary: 05C05 #Probability (math.PR) #Secondary: 60F05 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2104.02715
openalex publication_date 2021/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the additive functional Xn(α) 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 \Reα< 0, we prove that Xn(α), 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 \Reα> 0, for which Xn(α), 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 E[ξ] = 1 and 0 < Var [ξ] < ∞. 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 \Reα> \frac12, the limit random variable Y(α) can be expressed as a function of a normalized Brownian excursion.