THE COALESCENT POINT PROCESS OF BRANCHING TREES
THE COALESCENT POINT PROCESS OF BRANCHING TREES
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DOI：
10.1214/11-aap820
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发表时间：
2013-02-01
影响因子：
1.8
通讯作者：
Popovic, Lea
中科院分区：
文献类型：
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作者：
Lambert, Amaury;Popovic, Lea
我们定义了Bienayme-Galton-沃森（BOW）谱系的双无限单调标号.当前世代在时间上向后的谱系由聚结点过程（At; i >= 1）唯一地确定，其中A（i）是个体i和i + 1之间的聚结时间。有一个点测度的马尔可夫过程（δ（i）; i >= 1）保持跟踪更多的祖先关系，使得A（i）也是B-i的第一个点质量。这个点测度的过程也与平面BGW树中第h代中第一个存活粒子的谱系的非齐次脊分解密切相关，该树的条件是存活h代。分解涉及到一个点度量rho，它存储了脊柱右侧的子树数量。在适当的条件下，我们证明了这个点测度收敛到R+上的一个点测度与极限连续状态分支（CSB）过程。通过只考虑合并次数大于1/2的点对极限CSB布居进行离散，证明了合并点过程的相关不变性原理。极限合并点过程（B-i（1/2）; i >= 1）是高度过程在某一固定水平以下的偏移深度大于1/2的序列。在扩散的情况下，有没有多个祖先和（它是已知的）的聚结点过程是一个泊松点过程与明确的强度措施。本文证明了在一般情况下，重数合并过程（B-i（n）; i >= 1）是一个点质量的马氏链，并给出了它的转移函数的显式表达式，最后给出了它在离散情况下的两个应用.我们的结果表明，A（i）的序列是独立同分布的.当后代分布是线性分数时。同时，亚临界BOW过程的Yaglom准定态粒子数律也被瓦解。
We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson (BOW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process (At; i >= 1), where A(i) is the coalescence time between individuals i and i + 1. There is a Markov process of point measures (8(i); i >= 1) keeping track of more ancestral relationships, such that A(i) is also the first point mass of B-i.This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation h in a planar BGW tree conditioned to survive h generations. The decomposition involves a point measure rho storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on R+ associated with the limiting continuous-state branching (CSB) process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting CSB population by considering only points with coalescence times greater than epsilon.The limiting coalescent point process (B-i(epsilon); i >= 1) is the sequence of depths greater than epsilon of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. We prove that in the general case the coalescent process with multiplicities (B-i(epsilon); i >= 1) is a Markov chain of point masses and we give an explicit formula for its transition function.The paper ends with two applications in the discrete case. Our results show that the sequence of A(i) 's are i.i.d. when the offspring distribution is linear fractional. Also, the law of Yaglom's quasi-stationary population size for subcritical BOW processes is disintegrated with respect to the time to most recent common ancestor of the whole population.