## Advances in Applied Probability

### A weakly 1-stable distribution for the number of random records and cuttings in split trees

Cecilia Holmgren

#### Abstract

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

#### Article information

Source
Adv. in Appl. Probab., Volume 43, Number 1 (2011), 151-177.

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aap/1300198517

Digital Object Identifier
doi:10.1239/aap/1300198517

Mathematical Reviews number (MathSciNet)
MR2761152

Zentralblatt MATH identifier
1213.05037

#### Citation

Holmgren, Cecilia. A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. in Appl. Probab. 43 (2011), no. 1, 151--177. doi:10.1239/aap/1300198517. https://projecteuclid.org/euclid.aap/1300198517

#### References

• Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.
• Beljaev, Ju. K. and Maksimov, V. M. (1963). Analytical properties of a generating function for the number of renewals. Theoret. Prob. Appl. 8, 108–112.
• Bourdon, J. (2001). Size and path length of Patricia tries: dynamical sources context. Random Structures Algorithms 19, 289–315.
• Broutin, N. and Holmgren, C. (2011). The total path length of split trees. Submitted.
• Caliebe, A. (2003). Symmetric fixed points of a smoothing transformation. Adv. Appl. Prob. 35, 377–394.
• Caliebe, A. and Rösler, U. (2003). Fixed points with finite variance of a smoothing transformation. Stoch. Process. Appl. 107, 105–129.
• Devroye, L. (1998). Universal limit laws for depths in random trees. SIAM J. Comput. 28, 409–432.
• Devroye, L. (2005). Applications of Stein's method in the analysis of random binary search trees. In Stein's Method and Applications (Lecture Notes Ser. Inst. Math. Sci. Natl. Univ. Sing. 5), Singapore University Press, pp. 247–297.
• Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319–336.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
• Fill, J. A. and Janson, S. (2002). Quicksort asymptotics. J. Algorithms 44, 4–28.
• Fill, J. A. and Kapur, N. (2004). Limiting distributions for additive functionals on Catalan trees. Theoret. Comput. Sci. 326, 69–102.
• Flajolet, P., Roux, M. and Vallée, B. (2010). Digital trees and memoryless sources: from arithmetics to analysis. To appear in Discrete Math. Theoret. Comput. Sci.
• Gut, A. (1988). Stopped Random Walks. Springer, New York.
• Gut, A. (2005). Probability: A Graduate Course. Springer, New York.
• Holmgren, C. (2010). A weakly 1-stable limiting distribution for the number of random records and cuttings in split trees. Preprint. Available at http://arxiv.org/abs/1005.4590v1.
• Holmgren, C. (2010). Novel characteristics of split trees by use of renewal theory. Preprint. Available at http://arxiv.org/abs/1005.4594v1.
• Holmgren, C. (2010). Random records and cuttings in binary search trees. Combinatorics Prob. Comput. 19, 391–424.
• Iksanov, A. and Meiners, M. (2010). Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks. J. Appl. Prob. 47, 513–525.
• Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Prob. 12, 28–35.
• Janson, S. (2004). Random records and cuttings in complete binary trees. In Mathematics and Computer Science. III, Birkhäuser, Basel, pp. 241–253.
• Janson, S. (2006). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29, 139–179.
• Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
• Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
• Mahmoud, H. (1986). On the average internal path length of $m$-ary search trees. Acta Informatica 23, 111–117.
• Mahmoud, H. and Pittel, B. (1989). Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10, 52–75.
• Meir, A. and Moon, J. W. (1970). Cutting down random trees. J. Austral. Math. Soc. 11, 313–324.
• Neininger, R. and Rüschendorf, L. (1999). On the internal path length of $d$-dimensional quad trees. Random Structures Algorithms 15, 25–41.
• Panholzer, A. (2003). Non-crossing trees revisited: cutting down and spanning subtrees. In Discrete Random Walks (Paris, 2003), Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 265–276.
• Panholzer, A. (2006). Cutting down very simple trees. Quaest. Math. 29, 211–227.
• Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica 29, 238–261.