General Edgeworth expansions with applications to profiles of random trees

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  • Westfälische Wilhelms-Universität Münster
  • Faculty of Cybernetics, Taras Shevchenko National University of Kyiv


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 Ln:Z→R which converges in the mod-ϕ-sense. Applications to Stirling numbers of the first kind will be given in a separate paper.


Original languageEnglish
Pages (from-to)3478-3524
Number of pages47
JournalAnnals of Applied Probability
Issue number6
Publication statusPublished - 1 Dec 2017


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