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

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Listed:
  • Duffy, Ken R.
  • Meli, Gianfelice
  • Shneer, Seva

Abstract

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.

Suggested Citation

  • Duffy, Ken R. & Meli, Gianfelice & Shneer, Seva, 2019. "The variance of the average depth of a pure birth process converges to 7," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 88-93.
    • Handle: RePEc:eee:stapro:v:150:y:2019:i:c:p:88-93
    DOI: 10.1016/j.spl.2019.02.015
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    References listed on IDEAS

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    1. J. M. Marchingo & G. Prevedello & A. Kan & S. Heinzel & P. D. Hodgkin & K. R. Duffy, 2016. "T-cell stimuli independently sum to regulate an inherited clonal division fate," Nature Communications, Nature, vol. 7(1), pages 1-12, December.
    2. Biggins, J. D. & Grey, D. R., 1997. "A note on the growth of random trees," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 339-342, April.
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