2 Branching processes 2.1 Classi cation and extinction Informally, a branching process 10 is described as follows: let f p k g k 0, be a xed probability mass function. It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. the time since We also find the probability generating function of the Yaglom distribution of the process and rather explicit expressions for the transition rates for the so-called Q-process, that is the logistic branching process conditioned to stay alive into the indefinite future. It is well-known that a non-degenerate branching goes either extinction or explosion. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction. It is well-known that a non-degenerate branching goes either extinction or explosion. If P(no offspring)6= 0 there is a probability that the process will die out. the branching process. It is clear that the conditioned branching process has extinction probability one for any starting. We therefore condition the process to have n species today, we call that process the conditioned birth-death process (cBDP). The law of the total progeny of the tree is studied thanks to this correspondence. A population starts with a single ancestor who forms generation number 0.

Conditioning on very late extinction Irreducible branching process with P, M, ˆ, ˘ We want to work with a process which dies out almost surely: critical or subcritical process supercritical process with positive risk of extinction q, conditioned on extinction )subcritical process with eP, … Yaglom quasi-stationary limit (one-dimensional distribution conditional on non extinction) and the Q-process (process conditioned on non-extinction in the distant future). Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. the question arises whether (non-critical) general branching populations (also known as Crump-Mode-Jagers, or CMJ, processes) bound for extinction must behave like subcritical populations. function” Assume: (i) psame for all individuals (ii) individuals reproduce independently 4 Branching Processes Organise by generations: Discrete time. Then we show how to code the genealogy of a BGW tree thanks to a killed random walk. We answer this in the affirmative: a general, multi-type branching process conditioned to die out, remains a branching process, but one almost surely dying out. The distribution of the GWBP(ξ), conditioned on the extinction is the same as the distribution of the GWBP(ξˆ). Let X= number of offspring of an individual p(x) = P(X= x) = “offspring prob. In the end we will briefly state some more advanced results. It was showed in Li, 2000, Li, 2011 that a supercritical CB-process conditioned on extinction is equivalent to a subcritical one and a critical or subcritical CB-process conditioned on distant extinction time is equivalent to a special CBI-process. type but this would also be the case if the process were nontrivially critical, i.e. The age of the tree, i.e. Notice that the former is a priori a supercritical case, and that the latter is a subcritical case. When looking at phylogenies, we have a given number, say n, of extant taxa. It was showed in Li (2000, 2011) that a supercritical CB-process conditioned on extinction is equivalent to a subcritical one and a critical or subcritical CB-process conditioned on distant extinction time is equivalent to a special CBI-process. µ = 0 and the critical branching process (Aldous and Popovic, 2005; Popovic, 2004) where µ = λ.