A recursive distribution equation for the stable tree
We provide a new characterisation of Duquesne and Le Gall's $\alpha$-stable tree, $\alpha\in(1,2]$, as the solution of a recursive distribution equation (RDE) of the form $\mathcal{T}\overset{d}{=}g(\xi,\mathcal{T}_i, i\geq0)$, where $g$ is a concatenation operator, $\xi = (\xi_i, i\geq 0)$ a s...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Bernoulli Society for Mathematical Statistics and Probability
2024
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| _version_ | 1826315233447116800 |
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| author | Chee, N Rembart, F Winkel, M |
| author_facet | Chee, N Rembart, F Winkel, M |
| author_sort | Chee, N |
| collection | OXFORD |
| description | We provide a new characterisation of Duquesne and Le Gall's $\alpha$-stable
tree, $\alpha\in(1,2]$, as the solution of a recursive distribution equation
(RDE) of the form $\mathcal{T}\overset{d}{=}g(\xi,\mathcal{T}_i, i\geq0)$,
where $g$ is a concatenation operator, $\xi = (\xi_i, i\geq 0)$ a sequence of
scaling factors, $\mathcal{T}_i$, $i \geq 0$, and $\mathcal{T}$ are i.i.d.
trees independent of $\xi$. This generalises a version of the well-known
characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque
and Goldschmidt. By relating to previous results on a rather different class of
RDE, we explore the present RDE and obtain for a large class of similar RDEs
that the fixpoint is unique (up to multiplication by a constant) and
attractive. |
| first_indexed | 2024-03-07T08:08:05Z |
| format | Journal article |
| id | oxford-uuid:5db62e6a-62b4-4227-b02e-f507ee1f2cff |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:22:11Z |
| publishDate | 2024 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| record_format | dspace |
| spelling | oxford-uuid:5db62e6a-62b4-4227-b02e-f507ee1f2cff2024-11-14T13:15:19ZA recursive distribution equation for the stable treeJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5db62e6a-62b4-4227-b02e-f507ee1f2cffEnglishSymplectic ElementsBernoulli Society for Mathematical Statistics and Probability2024Chee, NRembart, FWinkel, MWe provide a new characterisation of Duquesne and Le Gall's $\alpha$-stable tree, $\alpha\in(1,2]$, as the solution of a recursive distribution equation (RDE) of the form $\mathcal{T}\overset{d}{=}g(\xi,\mathcal{T}_i, i\geq0)$, where $g$ is a concatenation operator, $\xi = (\xi_i, i\geq 0)$ a sequence of scaling factors, $\mathcal{T}_i$, $i \geq 0$, and $\mathcal{T}$ are i.i.d. trees independent of $\xi$. This generalises a version of the well-known characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque and Goldschmidt. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive. |
| spellingShingle | Chee, N Rembart, F Winkel, M A recursive distribution equation for the stable tree |
| title | A recursive distribution equation for the stable tree |
| title_full | A recursive distribution equation for the stable tree |
| title_fullStr | A recursive distribution equation for the stable tree |
| title_full_unstemmed | A recursive distribution equation for the stable tree |
| title_short | A recursive distribution equation for the stable tree |
| title_sort | recursive distribution equation for the stable tree |
| work_keys_str_mv | AT cheen arecursivedistributionequationforthestabletree AT rembartf arecursivedistributionequationforthestabletree AT winkelm arecursivedistributionequationforthestabletree AT cheen recursivedistributionequationforthestabletree AT rembartf recursivedistributionequationforthestabletree AT winkelm recursivedistributionequationforthestabletree |