Conditioned Galton–Watson trees do not grow
نویسنده
چکیده
A conditioned Galton–Watson tree is a random rooted tree that is (or has the same distribution as) the family tree of a Galton–Watson process with some given offspring distribution, conditioned on the total number of vertices. We let ξ be a random variable with the given offspring distribution; i.e., the number of offspring of each individual in the Galton–Watson process is a copy of ξ. We let ξ be fixed throughout the paper, and let Tn denote the corresponding conditioned Galton–Watson tree with n vertices. For simplicity, we consider only ξ such that P(ξ = 0) > 0 and P(ξ = 1) > 0; then Tn exists for all n ≥ 1. Furthermore, we assume that E ξ = 1 (the Galton–Watson process is critical) and σ := Var(ξ) <∞. The importance of this construction lies in that many combinatorially interesting random trees are of this type, for example the following:
منابع مشابه
Noncrossing trees are almost conditioned Galton-Watson trees
A non-crossing tree (NC-tree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this paper, we show that NC-trees with size n are conditioned Galton–Watson trees. As corollaries, we give the limit law of depth-first traversal processes and the limit profile of NC-trees.
متن کاملA Note on Conditioned Galton-watson Trees
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-W...
متن کاملThe lineage process in Galton-Watson trees and globally centered discrete snakes
We consider branching random walks built on Galton-Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, w...
متن کاملLocal limits of conditioned Galton-Watson trees: the infinite spine case
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limits of conditioned Galton-Watson trees. We then apply this condition to get new results in the critical case (with a general offspring distribution) and in the sub-critic...
متن کاملLocal Limits of Conditioned Galton-watson Trees I: the Infinite Spine Case
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limits of conditioned Galton-Watson trees. We then apply this condition to get new results in the critical case (with a general offspring distribution) and in the sub-critic...
متن کامل