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December 2017 General Edgeworth expansions with applications to profiles of random trees
Zakhar Kabluchko, Alexander Marynych, Henning Sulzbach
Ann. Appl. Probab. 27(6): 3478-3524 (December 2017). DOI: 10.1214/17-AAP1285

Abstract

We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16 (2006) 886–918], Fuchs, Hwang and Neininger [Algorithmica 46 (2006) 367–407], and Drmota and Hwang [Adv. in Appl. Probab. 37 (2005) 321–341]. The aforementioned results are special cases and corollaries of a general theorem: an Edgeworth expansion for an arbitrary sequence of random or deterministic functions $\mathbb{L}_{n}:\mathbb{Z}\to\mathbb{R}$ which converges in the mod-$\phi$-sense. Applications to Stirling numbers of the first kind will be given in a separate paper.

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Zakhar Kabluchko. Alexander Marynych. Henning Sulzbach. "General Edgeworth expansions with applications to profiles of random trees." Ann. Appl. Probab. 27 (6) 3478 - 3524, December 2017. https://doi.org/10.1214/17-AAP1285

Information

Received: 1 September 2016; Published: December 2017
First available in Project Euclid: 15 December 2017

zbMATH: 1382.60068
MathSciNet: MR3737930
Digital Object Identifier: 10.1214/17-AAP1285

Subjects:
Primary: 60G50
Secondary: 60F05 , 60F10 , 60F15 , 60J80 , 60J85

Keywords: Biggins martingale , Branching random walk , central limit theorem , Edgeworth expansion , Mode , mod-phi convergence , Profile , Random analytic function , Random tree , width

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 6 • December 2017
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