Galton-Watson树中的谱系过程与全局中心离散蛇
The lineage process in Galton--Watson trees and globally centered discrete snakes
摘要：
我们考虑分支随机游动建立在高尔顿-沃森树的后代分布有界的支持，条件有$n$节点，和他们的重新缩放收敛到布朗蛇。我们展示了一个概念的“全球中心离散蛇”，扩展了通常的设置中，位移应该为中心。我们证明了在一些附加的矩条件下，当$n$到$+\infty$时，“全局中心离散蛇”收敛到布朗蛇。证明依赖于一个精确的研究的血统的节点在高尔顿-沃森树的条件下的大小，和他们的联系与一个多项式过程[血统的节点$u$是矢量索引的$(k,j)$给出的数量的祖先的$u$有$k$的儿童和$u$是后代的$j$ th一个]。一些后果有关高尔顿-沃森树的大小的条件下也得来。
英文摘要：
We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node $u$ is the vector indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children and for which $u$ is a descendant of the $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived.
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