Convergence of the pruning processes of stable Galton-Watson trees
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911244434276352 |
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| author | Ojeda, Gabriel Berzunza Winter, Anita |
| author_facet | Ojeda, Gabriel Berzunza Winter, Anita |
| contents | Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue Lévy trees. Löhr, Voisin and Winter (2015) introduced the space of bi-measure $\mathbb{R}$-trees equipped with the so-called leaf sampling weak vague topology which allows them to unify the discrete and the continuous picture by considering them as instances of the same Feller-continuous Markov process with different initial conditions. Moreover, the authors show that these so-called pruning processes converge in the Skorokhod space of càdlàg paths with values in the space of bi-measure $\mathbb{R}$-trees, whenever the initial bi-measure $\mathbb{R}$-trees converge.
In this paper we provide an application to the above principle by verifying that a sequence of suitably rescaled critical conditioned Galton-Watson trees whose offspring distributions lie in the domain of attraction of a stable law of index $α\in (1,2]$ converge to the $α$-stable Lévy-tree in the leaf-sampling weak vague topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10489 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence of the pruning processes of stable Galton-Watson trees Ojeda, Gabriel Berzunza Winter, Anita Probability Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue Lévy trees. Löhr, Voisin and Winter (2015) introduced the space of bi-measure $\mathbb{R}$-trees equipped with the so-called leaf sampling weak vague topology which allows them to unify the discrete and the continuous picture by considering them as instances of the same Feller-continuous Markov process with different initial conditions. Moreover, the authors show that these so-called pruning processes converge in the Skorokhod space of càdlàg paths with values in the space of bi-measure $\mathbb{R}$-trees, whenever the initial bi-measure $\mathbb{R}$-trees converge. In this paper we provide an application to the above principle by verifying that a sequence of suitably rescaled critical conditioned Galton-Watson trees whose offspring distributions lie in the domain of attraction of a stable law of index $α\in (1,2]$ converge to the $α$-stable Lévy-tree in the leaf-sampling weak vague topology. |
| title | Convergence of the pruning processes of stable Galton-Watson trees |
| topic | Probability |
| url | https://arxiv.org/abs/2304.10489 |