Rotating random trees with Skorokhod's $M_1$ topology
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918367943720960 |
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| author | Aurillard, Antoine |
| author_facet | Aurillard, Antoine |
| contents | We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for $\mathbb R$-trees and Skorokhod's $M_1$ topology for càdlàg functions. As a first application, we study the $\mathbb R$-trees $\mathcal T_{x^{(α)}}$ encoded by excursions of spectrally positive $α$-stable Lévy processes for $α\in (1,2]$. In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an $α$-stable law with $α\in (1,2)$, and instead we prove that the scaling limit of the rotated trees is $\mathcal T_{x^{(α)}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rotating random trees with Skorokhod's $M_1$ topology Aurillard, Antoine Probability We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for $\mathbb R$-trees and Skorokhod's $M_1$ topology for càdlàg functions. As a first application, we study the $\mathbb R$-trees $\mathcal T_{x^{(α)}}$ encoded by excursions of spectrally positive $α$-stable Lévy processes for $α\in (1,2]$. In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an $α$-stable law with $α\in (1,2)$, and instead we prove that the scaling limit of the rotated trees is $\mathcal T_{x^{(α)}}$. |
| title | Rotating random trees with Skorokhod's $M_1$ topology |
| topic | Probability |
| url | https://arxiv.org/abs/2412.12023 |