Conditioned Galton–Watson Trees: The Shape Functional, and More on the Sum of Powers of Subtree Sizes and Its Mean

Abstract

AbstractFor a complex number $$\alpha $$ α , we consider the sum of the $$\alpha $$ α th powers of subtree sizes in Galton–Watson trees conditioned to be of size n. Limiting distributions of this functional $$X_n(\alpha )$$ X n ( α ) have been determined for $${\text {Re}}\alpha \ne 0$$ Re α ≠ 0 , revealing a transition between a complex normal limiting distribution for $${\text {Re}}\alpha < 0$$ Re α < 0 and a non-normal limiting distribution for $${\text {Re}}\alpha > 0$$ Re α > 0 . In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $${\text {Re}}\alpha = 0$$ Re α = 0 . The same results are also established in the case of the so-called shape functional $$X_n'(0)$$ X n ′ ( 0 ) , which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. In addition, we prove convergence of all moments in the case $${\text {Re}}\alpha < 0$$ Re α < 0 , where this result was previously missing, and establish new results about the asymptotic mean for real $$\alpha < 1/2$$ α < 1 / 2 .A novel feature for $${\text {Re}}\alpha =0$$ Re α = 0 is that we find joint convergence for several $$\alpha $$ α to independent limits, in contrast to the cases $${\text {Re}}\alpha \ne 0$$ Re α ≠ 0 , where the limit is known to be a continuous function of $$\alpha $$ α . Another difference from the case $${\text {Re}}\alpha \ne 0$$ Re α ≠ 0 is that there is a logarithmic factor in the asymptotic variance when $${\text {Re}}\alpha =0$$ Re α = 0 ; this holds also for the shape functional.The proofs are largely based on singularity analysis of generating functions.