Local limits of conditioned Galton-Watson trees: the infinite spine case

Abstract

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-critical cases (with a generic offspring distribution) on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.

Conditioning Galton–Watson Trees on Large Maximal Outdegree

“…In the present paper, we condition GW trees to have large maximal out-degree and use the framework in [1,2] to study local limits of large conditioned trees. We say that a probability distribution p = (p 0 , p 1 , p 2 , .…”

“…On the technical level, our proofs are extremely short and elementary, thanks in particular to the convenient framework in [1,2]. Nevertheless, we still would like to stress the following points: First, our new conditioning is the most natural way to get condensation tree in the limit, and this seems to be an intrinsic reason behind our short and elementary proofs; Second, potentially our new conditioning and some possible variants might be useful for studying condensation phenomenon in other settings; Finally, conditionings of GW trees seem more complete now, with conditioning on large height as one extreme case, our new conditioning as the opposite extreme case, and other conditionings in between.…”

“…Following the terminology of Galton-Watson processes, we also say that t is extinct if t ∈ T 0 , non-extinct otherwise. For precise definitions of all these notations in the present paragraph, refer to Section 2 in [1] and Section 2 in [2].…”

“…Janson [7] completed this result by proving that any sub-critical GW tree conditioned to have large total progeny converges to the condensation tree, if not to Kesten's tree. In [1,2], Abraham and Delmas provided a convenient framework to study local limits of conditioned GW trees, then they used this framework to prove essentially all previous results and some new ones. Specifically, in [1] they provided a criterion for local convergence of finite random trees to Kesten's tree, then gave short and elementary proofs of essentially all previous related results and some new ones.…”

“…In [1,2], Abraham and Delmas provided a convenient framework to study local limits of conditioned GW trees, then they used this framework to prove essentially all previous results and some new ones. Specifically, in [1] they provided a criterion for local convergence of finite random trees to Kesten's tree, then gave short and elementary proofs of essentially all previous related results and some new ones. In [2], they provided a criterion for local convergence of finite random trees to condensation tree, then generalized and completed essentially all previous related results by conditioning GW trees to have a large number of individuals with out-degree in a given set.…”