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On the asymptotic behaviour of random recursive trees in random environments

Part of: Graph theory Stochastic processes Limit theorems

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 

References

Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in random environment. Ann. Prob. 33, 645–673.Google Scholar
Babilot, M., Bougerol, P. and Élie, L. (1997). The random difference equation X n =A n X n-1 B n in the critical case. Ann. Prob. 25, 478–493.Google Scholar
Barabási, A. L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.Google Scholar
Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Prob. 22, 2152–2167.CrossRefGoogle Scholar
Biggins, J. D. and Grey, D. R. (1997). A note on the growth of random trees. Statist. Prob. Lett. 32, 339–342.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree of a scale-free random graph process. Random Structures Algorithms 18, 279–290.CrossRefGoogle Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.Google Scholar
Borovkov, K. A. and Motyer, A. (2005). On the asymptotic behavior of a simple growing point process model. Statist. Prob. Lett. 72, 265–275.Google Scholar
Devroye, L. (1988). Applications of the theory of records in the study of random trees. Acta Informatica 26, 123–130.Google Scholar
Doney, R. A. (1995). Spitzer's condition and ladder variables in random walks. Prob. Theory Relat. Fields 101, 577–580.Google Scholar
Élie, L. (1982). Compartement asymptotic du noyau potentiel sur les groupes de Lie. Ann. Sci. École Norm. Sup. 15, 257–364.Google Scholar
Hirano, K. (1998). Determination of the limiting coefficient for exponential functionals of random walks with positive drift. J. Math. Sci. Univ. Tokyo 5, 299–332.Google Scholar
Mahmoud, H. (1992). Evolution of Random Search Trees. John Wiley, New York.Google Scholar
Pittel, B. (1994). Note on the height of recursive trees and random m-ary search trees. Random Structures Algorithms 5, 337–347.Google Scholar
Rüschendorf, L. and Rachev, S. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770–799.Google Scholar
Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.CrossRefGoogle Scholar
Vatutin, V. A. and Dyakonova, E. E. (2004). Galton–Watson branching processes in random environment. II. Finite-dimensional distributions. Theory Prob. Appl. 49, 275–308.Google Scholar