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Asymptotic Joint Normality of Counts of Uncorrelated Motifs in Recursive Trees

Author

Listed:
  • Mohan Gopaladesikan

    (Purdue University)

  • Hosam Mahmoud

    (The George Washington University)

  • Mark Daniel Ward

    (Purdue University)

Abstract

We study the fringe of random recursive trees, by analyzing the joint distribution of the counts of uncorrelated motifs. Our approach allows for finite and countably infinite collections. To be able to deal with the collection when it is infinitely countable, we use measure-theoretic themes. Each member of a collection of motifs occurs a certain number of times on the fringe. We show that these numbers, under appropriate normalization, have a limiting joint multivariate normal distribution. We give a complete characterization of the asymptotic covariance matrix. The methods of proof include contraction in a metric space of distribution functions to a fixed-point solution (limit distribution). We discuss two examples: the finite collection of all possible motifs of size four, and the infinite collection of rooted stars. We conclude with remarks to compare fringe-analysis with matching motifs everywhere in the tree.

Suggested Citation

  • Mohan Gopaladesikan & Hosam Mahmoud & Mark Daniel Ward, 2014. "Asymptotic Joint Normality of Counts of Uncorrelated Motifs in Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 863-884, December.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:4:d:10.1007_s11009-013-9333-z
    DOI: 10.1007/s11009-013-9333-z
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    Cited by:

    1. Panpan Zhang & Hosam M. Mahmoud, 2020. "On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 625-645, June.
    2. Jeffrey Gaither & Hosam Mahmoud & Mark Daniel Ward, 2017. "On the Variety of Shapes in Digital Trees," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1225-1254, December.

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