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A central limit theorem for additive functionals of increasing trees

Part of: Combinatorial probability Graph theory Limit theorems

Published online by Cambridge University Press:  08 March 2019

Dimbinaina Ralaivaosaona  and
Stephan Wagner
Dimbinaina Ralaivaosaona
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa
Stephan Wagner*
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa
*
*Corresponding author. Email: swagner@sun.ac.za
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Abstract

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Special Issue 4: Analysis of Algorithms , July 2019 , pp. 618 - 637
Copyright
© Cambridge University Press 2019 

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Footnotes

†

An extended abstract of this paper was presented at the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Kraków, 4–8 July 2016: see [15].

§

Supported by the Division for Research Development (DRD) of Stellenbosch University.

This material is based upon work supported by the National Research Foundation under grant number 96236.

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