March 2013 Asymptotic analysis of Hoppe trees
Kevin Leckey, Ralph Neininger
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J. Appl. Probab. 50(1): 228-238 (March 2013). DOI: 10.1239/jap/1363784435


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.


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Kevin Leckey. Ralph Neininger. "Asymptotic analysis of Hoppe trees." J. Appl. Probab. 50 (1) 228 - 238, March 2013.


Published: March 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1264.60021
MathSciNet: MR3076783
Digital Object Identifier: 10.1239/jap/1363784435

Primary: 60C05 , 60F05
Secondary: 60G42 , 68R05

Keywords: combinatorial probability , Hoppe urn , martingale , Random tree , weak convergence

Rights: Copyright © 2013 Applied Probability Trust


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Vol.50 • No. 1 • March 2013
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