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Using Stein's method to show Poisson and normal limit laws for fringe subtrees.
Stockholm University, Faculty of Science, Department of Mathematics.
Uppsala universitet.
2014 (English)In: Discrete mathematics and theoretical computer science (Online), ISSN 1462-7264, E-ISSN 1365-8050, Vol. Proc., no BA, 169-180 p.Article in journal (Refereed) Published
Abstract [en]

The use of Stein’s method and certain couplings allow provision of simple proofs showing that in both of these trees, the number of fringe subtrees of size k < n, where k ! 1, can be approximated by a Poisson distribution. Combining these results and another version of Stein’s method, we can also show that for k = o(pn), the number of fringe subtrees in both types of random trees has asymptotically a normal distribution as n ! 1. Furthermore, using the Cram´er–Wold device, we show that a random vector with components corresponding to the random number of copies of certain fixed fringe subtrees Ti, has asymptotically a multivariate normal distribution. We can then use these general results on fringe subtrees to obtain simple solutions to a broad range of problems relating to random trees; as an example, we can prove that the number of protected nodes in the binary search tree has asymptotically a normal distribution.

Place, publisher, year, edition, pages
2014. Vol. Proc., no BA, 169-180 p.
Keyword [en]
Fringe subtrees. Stein’s method, Couplings, Limit laws, Binary search trees, Recursive trees
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-114311OAI: oai:DiVA.org:su-114311DiVA: diva2:791141
Available from: 2015-02-26 Created: 2015-02-26 Last updated: 2017-12-04Bibliographically approved

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