Fringe subtrees of split trees and fractional split trees
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914414371799040 |
|---|---|
| author | Holmgren, Cecilia Ischebeck, Jasper Janson, Svante |
| author_facet | Holmgren, Cecilia Ischebeck, Jasper Janson, Svante |
| contents | We consider additive functionals $X_n(ϕ)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fringe subtrees of split trees and fractional split trees Holmgren, Cecilia Ischebeck, Jasper Janson, Svante Probability 60C05 (Primary) 60F05 (Secondary) We consider additive functionals $X_n(ϕ)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method. |
| title | Fringe subtrees of split trees and fractional split trees |
| topic | Probability 60C05 (Primary) 60F05 (Secondary) |
| url | https://arxiv.org/abs/2603.21780 |