On the profile of trees with a given degree sequence
| Autor: | Angtuncio, Osvaldo, Bravo, Gerónimo Uribe |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{{\bf s}}$ on rooted plane trees whose degree sequence equals ${\bf s}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (1997) and Kersting (2011). A technical result is needed to ensure that trees with law $\mathbb{P}_{{\bf s}}$ have enough individuals in the first generations, and this is handled through novel path transformations and fluctuation theory of exchangeable increment processes. As a consequence, we obtain a boundedness criterion for the inhomogeneous continuum random tree introduced by Aldous, Miermont and Pitman (2004). Comment: 32 pages, 7 figures |
| Databáze: | arXiv |
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