A Central Limit Theorem for Almost Local Additive Tree Functionals

Abstract

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are "almost local" in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction process that was studied

On general subtrees of a conditioned Galton–Watson tree

“…It is possible that this may be overcome by truncations and some variance estimates, but again more work is needed. (The extension in [19] applies to the case when t is a star with root degree ∆ (including Example 4.2 with ∆ = 2) and E ξ 2∆+1 < ∞; this might suggest further extensions.) This problem is thus left for future research.…”

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

“…Among these, we mention the total path length defined as the sum of the distances to the root of all vertices, the Wiener index [43] defined as the sum of the distances between all pairs of vertices, the shape functional, the Sackin index, the Colless index and the cophenetic index, see [42] for their definitions and also [14] for their representation using additive functionals, and the references therein. See also [39] for other functionals such that the number of matchings, dominating sets, independent sets for trees. We also mention the Shao and Sokal's B 1 index [6,42] defined by…”

“…These being almost independent, we understand intuitively why the limit distribution is Gaussian. See [29,39,45] for asymptotic results in the local regime. In the global regime, the toll function is large when the subtree is large; so the main contribution comes from large subtrees which are strongly dependent.…”

On general subtrees of a conditioned Galton-Watson tree

“…It is possible that this may be overcome by truncations and some variance estimates, but again more work is needed. (The extension in [18] applies to the case when t is a star with root degree ∆ (including Example 4.2 with ∆ = 2) and E ξ 2∆+1 < ∞; this might suggest further extensions.) This problem is thus left for future research.…”