arXiv 0704.30982007-04math.PR

分裂树的等高线是一个Lévy过程

The contour of splitting trees is a Lévy process

Amaury Lambert

摘要：

分裂树是随机树，其中个体在一生中以恒定的速率生育，具有一般分布，即i.i.d.。他们自己的副本。一个分裂树的宽度过程是一个二元的，同质的Crump-Mode-Jagers（CMJ）过程，除非寿命分布是指数分布，否则不是马尔可夫过程。在这里，我们允许出生率是无限的，也就是说，对出生时间和寿命的新生儿形成一个泊松点过程沿其母亲的一生，可能具有无限的强度措施。一棵分裂树是一棵随机的（所谓的）时间顺序树。时序树的每个元素都是离散树中某个个体$v$（顶点）的（所谓的）存在点$(v,τ)$，其中$τ$是称为时序层（时间）的非负实数。我们引入了存在点的全序，称为线性序，以及从树到实数线的映射$φ$，它保持了这个顺序。$φ$的逆称为探索过程，该逆在时间顺序上的投影称为轮廓过程。对于截断到水平$τ$的分裂树，我们证明了这样定义的轮廓过程是反映在$τ$之下的Lévy过程，并且在击中0时被杀死。这允许导出属性（一）分裂树：概念上的证明Le Gall-Le Jan定理在有限变分的情况下，例外点，合并点过程，年龄分布;（二）CMJ过程：一维边缘，条件，极限定理，渐近数的个人与无限与有限的后裔。

英文摘要：

Splitting trees are those random trees where individuals give birth at constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump--Mode--Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential. Here, we allow the birth rate to be infinite, that is, pairs of birth times and lifespans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point $(v,τ)$ of some individual $v$ (vertex) in a discrete tree, where $τ$ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping $φ$ from the tree into the real line which preserves this order. The inverse of $φ$ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level $τ$, we prove that thus defined contour process is a Lévy process reflected below $τ$ and killed upon hitting 0. This allows to derive properties of (i) splitting trees: conceptual proof of Le Gall--Le Jan's theorem in the finite variation case, exceptional points, coalescent point process, age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems, asymptotic numbers of individuals with infinite vs finite descendances.

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摘要：

英文摘要：

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