论文标题
从条件稳定的Galton-Watson树中得出的破碎过程的不变性原理
Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees
论文作者
论文摘要
Aldous，Evans和Pitman（1998）研究了碎片过程的行为，这些过程是从删除$ n $标记的顶点上均匀随机树的边缘得出的。他们特别表明，经过适当的重新进行了重新进行，上述碎片化过程将$ n \ rightarrow \ infty $融合到通过以泊松方式沿其骨骼切割布朗尼CRT获得的布朗CRT的破碎过程。 在这项工作中，我们继续进行上述调查并研究通过从关键的Galton-Watson树$ \ Mathbf {T} _ {N} $中删除随机选择的边缘获得的分裂过程，其条件是具有$ n $ dertices，其offspring属于稳定的属性属于Index $ aindex $ aindex $α\α\ n $ n $ n $ n $ n $ n $ n.2]的属性。 fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $α$-stable Lévy tree of index $α\in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the Miermont（2001）研究的$α$稳定的Lévy游览，这扩展了Bertoin（2000）的结果。
Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n \rightarrow \infty$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner. In this work, we continue the above investigation and study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $\mathbf{t}_{n}$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $α\in (1,2]$. Our main results establish that, after rescaling, the fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $α$-stable Lévy tree of index $α\in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized $α$-stable Lévy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.
