The Annals of Applied Probability

General Edgeworth expansions with applications to profiles of random trees

Zakhar Kabluchko, Alexander Marynych, and Henning Sulzbach

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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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3478-3524.

Dates
Received: September 2016
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328706

Digital Object Identifier
doi:10.1214/17-AAP1285

Mathematical Reviews number (MathSciNet)
MR3737930

Zentralblatt MATH identifier
1382.60068

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60F10: Large deviations 60F15: Strong theorems

Keywords
Biggins martingale branching random walk central limit theorem Edgeworth expansion mod-phi convergence mode profile random analytic function random tree width

Citation

Kabluchko, Zakhar; Marynych, Alexander; Sulzbach, Henning. General Edgeworth expansions with applications to profiles of random trees. Ann. Appl. Probab. 27 (2017), no. 6, 3478--3524. doi:10.1214/17-AAP1285. https://projecteuclid.org/euclid.aoap/1513328706


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