HAL : hal-00672041, version 1
 arXiv : 1112.0256
 Limit distributions for multitype branching processes of m-ary search trees
 (01/12/2011)
 A particular continuous-time multitype branching process is considered, it is the continuous-time embedding of a discrete-time process which is very popular in theoretical computer science: the m-ary search tree (m is an integer). There is a well-known phase transition: when m \leq 26, the asymptotic behavior of the process is Gaussian, but for m \geq 27 it is no more Gaussian and a limit W of a complex-valued martingale arises. Thanks to the branching property it appears as a solution of a smoothing equation of the type Z = e^{-{\lambda}T}(Z(1) + ... + Z(m)), where {\lambda} \in C, the Z(k) are independent copies of Z and T is a R_+-valued random variable, independent of the Z(k). This distributional equation is extensively studied by various approaches. The existence and unicity of solution of the equation are proved by contraction methods. The fact that the distribution of W is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of W is obtained by considering W as the limit of a complex Mandelbrot cascade.
 1 : Laboratoire de Mathématiques de Versailles (LM-Versailles) CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines 2 : Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885 (LMAM) Université de Bretagne Sud
 Domaine : Mathématiques/Probabilités
 Lien vers le texte intégral : http://fr.arXiv.org/abs/1112.0256
 hal-00672041, version 1 http://hal.archives-ouvertes.fr/hal-00672041 oai:hal.archives-ouvertes.fr:hal-00672041 Contributeur : Nadège Arnaud <> Soumis le : Lundi 20 Février 2012, 14:08:41 Dernière modification le : Lundi 20 Février 2012, 14:08:41