Limiting Behavior Of Additive Functionals On The Stable Tree

Abstract

We consider the normalized stable tree T with index γ ∈ (1, 2] and we study the asymptotic behavior of additive functionals of the formwhere µ is the mass measure on T , H(x) is the height of x and σ r,x (resp. h r,x ) is the mass (resp. height) of the subtree of T above level r containing x. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees [2]. We distinguish different regimes depending on the behavior of β/α 1−1/γ and describe the limiting random variable in terms of the total height h of T and an independent (1 − 1/γ)-stable subordinator.