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 clear that the conditioned branching process has extinction probability one for any starting. 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 branching process. 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. µ = 0 and the critical branching process (Aldous and Popovic, 2005; Popovic, 2004) where µ = λ. Let X= number of offspring of an individual p(x) = P(X= x) = “offspring prob. Then we show how to code the genealogy of a BGW tree thanks to a killed random walk. The distribution of the GWBP(ξ), conditioned on the extinction is the same as the distribution of the GWBP(ξˆ). If P(no offspring)6= 0 there is a probability that the process will die out. function” Assume: (i) psame for all individuals (ii) individuals reproduce independently 4 Branching Processes Organise by generations: Discrete time. Notice that the former is a priori a supercritical case, and that the latter is a subcritical case. The law of the total progeny of the tree is studied thanks to this correspondence. 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 non-degenerate branching goes either extinction or explosion. 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. 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. A population starts with a single ancestor who forms generation number 0. The age of the tree, i.e. 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. We therefore condition the process to have n species today, we call that process the conditioned birth-death process (cBDP). type but this would also be the case if the process were nontrivially critical, i.e. It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out.

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. When looking at phylogenies, we have a given number, say n, of extant taxa. 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. 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, … the time since Yaglom quasi-stationary limit (one-dimensional distribution conditional on non extinction) and the Q-process (process conditioned on non-extinction in the distant future). In the end we will brieﬂy state some more advanced results.

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