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Asymptotic Analysis of Hoppe Trees

Part of: Combinatorial probability Stochastic processes Discrete mathematics in relation to computer science Limit theorems

Published online by Cambridge University Press:  30 January 2018

Kevin Leckey  and
Ralph Neininger
Kevin Leckey*
Affiliation:
Goethe University Frankfurt
Ralph Neininger*
Affiliation:
Goethe University Frankfurt
*
∗ Postal address: Institute for Mathematics, Goethe University, 60054 Frankfurt am Main, Germany.
∗ Postal address: Institute for Mathematics, Goethe University, 60054 Frankfurt am Main, Germany.
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Abstract

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We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

Keywords

Hoppe urnrandom treeweak convergencemartingalecombinatorial probability

MSC classification

Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 60G42: Martingales with discrete parameter 68R05: Combinatorics
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 228 - 238
Copyright
Copyright © Applied Probability Trust 2013 

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