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Branching random walks on binary search trees: convergence of the occupation measure

Published online by Cambridge University Press:  29 October 2010

Eric Fekete
Eric Fekete*
Affiliation:
UVSQ, Département de Mathématiques, 45 av. des États-Unis, 78035 Versailles Cedex, France
*
Corresponding author: [email protected]
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Abstract

We consider branching random walks with binary search trees as underlying trees. We show that the occupation measure of the branching random walk, up to some scaling factors, converges weakly to a deterministic measure. The limit depends on the stable law whose domain of attraction contains the law of the increments. The existence of such stable law is our fundamental hypothesis. As a consequence, using a one-to-one correspondence between binary trees and plane trees, we give a description of the asymptotics of the profile of recursive trees. The main result is also applied to the study of the size of the fragments of some homogeneous fragmentations.

Keywords

Random binary search treebranching random walkoccupation measurefragmentationrecursive tree
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 286 - 298
Copyright
© EDP Sciences, SMAI, 2010

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References

Aldous, D., Tree-based models for random distribution mass. J. Statist. Phys. 73 (1993) 625–641. CrossRef
Bertoin, J., The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. 5 (2003) 395–416. CrossRef
Biggins, J.D., Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25–37. CrossRef
P. Billingsley, Probability and measure. Second edition. John Wiley & Sons, New York (1986).
L. Breiman, Probability. Second edition. SIAM (1992).
Brown, G.G. and Shubert, B.O., On random binary trees. Math. Oper. Res. 9 (1985) 43–65. CrossRef
Chassaing, P. and Schaeffer, G., Random planar lattices and integrated superbrownian excursion. Probab. Theory Relat. Fields 128 (2004) 161–212. CrossRef
Chauvin, B., Klein, T., Marckert, J.F. and Rouault, A., Martingales and profile of binary search trees. Electron. J. Probab. 10 (2005) 420–435. CrossRef
Devroye, L. and Hwang, H.K., Width and more of the profile for random trees of logarithmic height. Ann. Appl. Probab. 16 (2006) 886–918. CrossRef
Drmota, M., Profile and height of random binary search trees. J. Iranian Stat. Soc. 3 (2004) 117–138.
M. Fuchs, H.-K. Hwang and R. Neininger, Profiles of random trees: limit theorems for random recursive trees and binary search trees. Available at: http://algo.stat.sinica.edu.tw (2005).
Janson, S. and Marckert, J.F., Convergence of discrete snakes. J. Theory Probab. 18 (2005) 615–645. CrossRef
O. Kallenberg, Fundations of Modern Probability. Second edition. Springer-Verlag, New York (2001).
D.E. Knuth, The art of computer programing, Volume 1: Fundamental algorithms. Second edition. Addison-Wesley, Reading, MA (1997).
M. Kuba and A. Panholzer, The left-right-imbalance of binary search trees. Available at: http://info.tuwien.ac.at/panholzer (2006).
Louchard, G., Exact and asymptotic distributions in digital and binary search trees. RAIRO Theoret. Inform. Appl. 21 (1987) 479–496. CrossRef
H. Mahmoud, Evolution of Random Search Trees. John Wiley, New York (1992).
Mahmoud, H.M. and Neininger, R., Distribution of distances in random binary search trees. Ann. Appl. Prob. 13 (2003) 253–276.
Mahmoud, H.M. and Smythe, R.T., A survey of recursive trees. Theoret. Probab. Math. Statist. 51 (1995) 1–27.
Marckert, J.-F., The rotation correspondence is asymptotically a dilatation. Random Struct. Algorithms 24 (2004) 118–132. CrossRef
Slade, G. and Hara, T., The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-brownian excursion. J. Math. Phys. 41 (2000) 1244–1293.