The Genealogy of Conditioned Critical Galton--Watson Trees

Abstract

Let $T$ be the random family tree associated with the critical Galton--Watson process $\{Z_{n}\}_{n\geq0}$. Geiger (1999) provided an explicit representation of the law of $T$ conditioned on $\{Z_{n}>0\}$ by inductively constructing the conditioned tree $\tilde{T}_n$ along the line of descent of the left--most particle. Intuitively, $\tilde{T}_n$ is composed of a ``spine'', with a conditioned GW tree attached to each of its ``left'' siblings and an ordinary GW tree attached to each of its ``right'' siblings. The structure of the ``right'' subtree is partially discussed in Geiger (1999), whereas the information about the ``left'' subtree remains largely unexplored. In this paper we investigate the genealogy of the ``left'' and ``right'' subtrees of $\tilde{T}_n$, and some intrinsic properties will be specified, including the branching mechanism, the limiting behavior of the coalescent times, the limiting number of siblings of the spine, and the scaling limit distribution of the particles. We prove that the sequence of coalescent times of the ``right'' subtree converge in distribution to a sequence of ``nested uniform random variables'' by scaling, interestingly, so do the sequence of coalescent times for the ``left'' subtree after a functional transformation. We also establish the asymptotic independence of the two sequences of coalescent times, which in turn implies the asymptotic independence of the ``left'' and ``right'' subtrees. Finally, as an application, we present a probabilistic proof of the conditional limit theorem for the critical GW process established by Spitzer (unpublished) and Lamperti and Ney (1968), that is, for any fixed $0<t<1$, $Z_{[nt]}/n$ converges in distribution to the sum of two independent exponential random variables with different parameters, which exactly come from the ``left'' and ``right'' subtrees respectively.