The stable trees revisited
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912893108224000 |
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| author | Goldschmidt, Christina Hill, Liam |
| author_facet | Goldschmidt, Christina Hill, Liam |
| contents | We introduce a new, relatively simple, line-breaking construction of the $α$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(α-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process.
We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaymé branching process with critical offspring distribution in the domain of attraction of an $α$-stable law (for $α\in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $α$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The stable trees revisited Goldschmidt, Christina Hill, Liam Probability 60C05, 60J80, 60G52, 05C05 We introduce a new, relatively simple, line-breaking construction of the $α$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(α-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaymé branching process with critical offspring distribution in the domain of attraction of an $α$-stable law (for $α\in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $α$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction. |
| title | The stable trees revisited |
| topic | Probability 60C05, 60J80, 60G52, 05C05 |
| url | https://arxiv.org/abs/2512.17533 |