Asymptotic genealogy of a critical branching process

Abstract

Consider a continuous-time binary branching process conditioned to have population size n at some time t, and with a chance p for recording each extinct individual in the process. Within the family tree of this process, we consider the smallest subtree containing the genealogy of the extant individuals together with the genealogy of the recorded extinct individuals. We introduce a novel representation of such subtrees in terms of a point-process, and provide asymptotic results on the distribution of this point-process as the number of extant individuals increases. We motivate the study within the scope of a coherent analysis for an a priori model for macroevolution.Comment: 30 page

The conditioned reconstructed process

“…This has already been established in a different way for λ = 1 in Aldous and Popovic (2005);Popovic (2004).…”

“…In the following, time 0 is today and t or the origin of the tree, so time is increasing going into the past. Special cases of the birth-death process are the Yule model (Yule, 1924) where µ = 0 and the critical branching process (Aldous and Popovic, 2005;Popovic, 2004) where µ = λ. When looking at phylogenies, we have a given number, say n, of extant taxa.…”

“…This had been done for the critical branching process in Aldous and Popovic (2005);Popovic (2004). In Section 3, we calculate the probability distribution of the age of a given tree on n species, assuming a uniform prior on (0, ∞) for the age of the tree.…”

“…We do that using a point process representation. The following point process has first been considered in connection with trees in Aldous and Popovic (2005);Popovic (2004). Definition 2.1.…”

New Analytic Results for Speciation Times in Neutral Models

“…The most popular neutral model is the so-called Yule model [2,6,25] Recently, a critical branching process as a neutral model for speciation was introduced [1,17]. In the critical branching process, each species has an exponential (rate λ) lifetime during which it produces offspring according to a Poisson (rate λ) process.…”