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On the asymptotic behaviour of random recursive trees in random environments

Part of: Limit theorems Graph theory Stochastic processes

Published online by Cambridge University Press:  08 September 2016

K. A. Borovkov  and
V. A. Vatutin
K. A. Borovkov*
Affiliation:
The University of Melbourne
V. A. Vatutin*
Affiliation:
Steklov Mathematical Institute
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: kostya@ms.unimelb.edu.au
∗∗ Postal address: Steklov Mathematical Institute RAS, Gubkin St. 8, 119991 Moscow, Russia.
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Abstract

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We consider growing random recursive trees in random environments, in which at each step a new vertex is attached (by an edge of random length) to an existing tree vertex according to a probability distribution that assigns the tree vertices masses proportional to their random weights. The main aim of the paper is to study the asymptotic behaviour of the distance from the newly inserted vertex to the tree's root and that of the mean numbers of outgoing vertices as the number of steps tends to ∞. Most of the results are obtained under the assumption that the random weights have a product form with independent, identically distributed factors.

Keywords

Random recursive treerandom environmentSpitzer's conditiondistance to the rootoutdegree

MSC classification

Primary: 05C80: Random graphs
Secondary: 60G50: Sums of independent random variables; random walks 05C05: Trees 60F99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 1047 - 1070
Copyright
Copyright © Applied Probability Trust 2006 

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