Asymptotic Joint Normality of Counts of Uncorrelated Motifs in Recursive Trees

Mohan Gopaladesikan (), Hosam Mahmoud () and Mark Daniel Ward ()
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Mohan Gopaladesikan: Purdue University
Hosam Mahmoud: The George Washington University
Mark Daniel Ward: Purdue University

Methodology and Computing in Applied Probability, 2014, vol. 16, issue 4, 863-884

Abstract: 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.

Keywords: Analysis of algorithms; Random trees; Recurrence; Motif; Contraction; MSC 05C05; MSC 60C05; MSC 68P10; MSC 68W40; MSC 11B37 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-013-9333-z

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