Scaling limits of random Pólya trees

Scaling limits of random Pólya trees Pólya trees are rooted trees considered up to symmetry. We establish the convergence of large uniform random Pólya trees with arbitrary degree restrictions to Aldous’ Continuum Random Tree with respect to the Gromov–Hausdorff metric. Our proof is short and elementary, and it is based on a novel decomposition: it shows that the global shape of a random Pólya tree is essentially dictated by a large Galton–Watson tree that it contains. We also derive sub-Gaussian tail bounds for both the height and the width, which are optimal up to constant factors in the exponent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

Scaling limits of random Pólya trees

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0770-4
Publisher site
See Article on Publisher Site

Abstract

Pólya trees are rooted trees considered up to symmetry. We establish the convergence of large uniform random Pólya trees with arbitrary degree restrictions to Aldous’ Continuum Random Tree with respect to the Gromov–Hausdorff metric. Our proof is short and elementary, and it is based on a novel decomposition: it shows that the global shape of a random Pólya tree is essentially dictated by a large Galton–Watson tree that it contains. We also derive sub-Gaussian tail bounds for both the height and the width, which are optimal up to constant factors in the exponent.

Journal

Probability Theory and Related FieldsSpringer Journals

Published: Mar 9, 2017

References

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