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Arbres, Processus de branchement non markoviens et Processus de Lévy

Abstract : In this dissertation, we focus on three developments of splitting trees introduced by Geiger & Kersting (1997), and on Crump-Mode-Jagers (CMJ) branching processes associated with them. These random trees model a population where all individuals have i.i.d. lifelengths and during their lives, they give birth at constant rate b to copies of themselves. The process counting the number of extant individuals through time is a binary and homogeneous CMJ process, that can be seen as a generalization of the Markovian birth and death process in which lifetimes are exponentially distributed. First, we consider a mainland-island model, which generalizes that of Karlin and McGregor and in which individuals carrying types immigrate at rate T to an island and start families that evolve independently and according to the previously described mechanism. Various assumptions are made about how types are chosen (either each new immigrant is of a different type, or it is of type i with probability pi, etc.) and we determine the asymptotic relative abundances of each type in the total population. In the "new immigrant = new type" case, the limiting distribution follows a GEM distribution with parameter T/b and we note that it only depends on this ratio and not on the lifelength distribution. Second, we study another population model where mutations can occur at birth of individuals with a certain probability. We consider a model with infinitely many alleles, that is, each new mutant is of a type (or allele) never encountered before, and neutral because individuals all behave in the same way regardless of their types. We study the allelic partition of the population by looking at its frequency spectrum which describes the number of types of a given age and carried by a given number of individuals. Using random characteristics techniques, we obtain results about its asymptotic behavior. We also prove the distribution convergence of sizes of largest families and of ages of the oldest ones. In the last chapter, we focus on spectrally positive Lévy processes that do not drift to infinity and that we condition to stay positive in a new way. A process X starting from x > 0 is conditioned to reach arbitrarily large heights before hitting 0 where the term height has to be understood in the meaning of the height process of Duquesne & Le Gall (2002). The law of the conditioned process is defined with an h-transform via a martingale. When X has finite variation, the main argument is that X can be seen as the contour process of a splitting tree and thus to condition the Lévy process is equivalent to condition the tree to reach arbitrarily large generations. When X has infinite variation, the height process is defined via local times and the martingale is constructed from the exploration process defined by Duquesne and Le Gall.
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Contributor : Mathieu Richard <>
Submitted on : Wednesday, December 7, 2011 - 2:37:45 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:34 PM
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  • HAL Id : tel-00649235, version 1


Mathieu Richard. Arbres, Processus de branchement non markoviens et Processus de Lévy. Probabilités [math.PR]. Université Pierre et Marie Curie - Paris VI, 2011. Français. ⟨tel-00649235⟩



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