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Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees

Authors Dimbinaina Ralaivaosaona, Matas Sileikis, Stephan Wagner

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Trees
Keywords
  • Galton-Watson trees
  • central limit theorem
  • additive functional

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Abstract

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger.

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Dimbinaina Ralaivaosaona, Matas Sileikis, and Stephan Wagner. Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.33

Author Details

Dimbinaina Ralaivaosaona
  • Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa
Matas Sileikis
  • The Czech Academy of Sciences, Institute of Computer Science, Pod Vodárenskou věží 2, 182 07 Prague, Czech Republic. With institutional support RVO:67985807.
Stephan Wagner
  • Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa

Funding

  • Ralaivaosaona, Dimbinaina: The first author was partially supported by the Division for Research Development (DRD) of Stellenbosch University.
  • Sileikis, Matas: The second author was supported by the Czech Science Foundation, grant number GJ16-07822Y.
  • Wagner, Stephan: The third author is supported by the National Research Foundation of South Africa under grant number 96236.

References

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