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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2010.07880 |
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| _version_ | 1866915471110963200 |
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| author | Ojeda, Gabriel Berzunza Holmgren, Cecilia |
| author_facet | Ojeda, Gabriel Berzunza Holmgren, Cecilia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_07880 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees Ojeda, Gabriel Berzunza Holmgren, Cecilia Probability 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. |
| title | Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees |
| topic | Probability |
| url | https://arxiv.org/abs/2010.07880 |