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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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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.

Cite AsGet BibTex

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

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.

Subject Classification

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

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References

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