The variance of the average depth of a pure birth process converges to 7

If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly in time. In this letter, we instead consider the variance of...

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Bibliographic Details
Published inarXiv.org
Main Authors Duffy, Ken R, Meli, Gianfelice, Shneer, Seva
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 01.03.2019
Subjects
Childbirth & labor
Convergence
Mathematics - Probability
Trees
Variations
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Summary:If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly in time. In this letter, we instead consider the variance of the average depth of leaves within each individual tree, establishing that, in contrast, it converges to a constant, \(7\). This result indicates that while the variance in leaf depths amongst the ensemble of pure birth processes undergoes large fluctuations, the average depth across individual trees is much more consistent.
ISSN:2331-8422
DOI:10.48550/arxiv.1812.05966