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

Ken R. Duffy, Gianfelice Meli, Seva Shneer

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
41 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)88-93
Number of pages6
JournalStatistics and Probability Letters
Early online date5 Mar 2019
Publication statusPublished - Jul 2019


  • Pure birth process
  • Variance of the average depth

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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