Geodesic
Almost sure asymptotic expansions for profiles of simply generated random trees
Teoriâ slučajnyh processov, Tome 24 (2019) no. 1, pp. 49-63

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This paper is a continuation of the analysis of Edgeworth expansions for one-split branching random walk and its application to random trees. We provide new results for profile, mode and width for several simply generated random trees, in particular for random recursive trees, $p$-oriented recursive trees and $D$-ary random trees. Our results are corollaries of a general Edgeworth expansion for a one-split branching random walk proved by Kabluchko, Marynych and Sulzbach [The Annals of Applied Probability 27(6): 3478–3524, 2017]. We derive an additional characterization of the random variables appearing in the coefficients of the asymptotic expansions by calculating explicitly corresponding fixed-point equations of a branching type. We further provide numerical simulations justifying our theoretical findings.
Keywords: binary search tree, branching random walk, $D$-ary tree, Edgeworth expansion, fixed-point equation, one-split branching random walk, $p$-oriented recursive tree, profile, random analytic function, random recursive tree, random tree, width.
Mots-clés : mode, PORT, simulation 
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     author = {Vladyslav Bogun},
     title = {Almost sure asymptotic expansions for profiles of simply generated random trees},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {49--63},
     publisher = {mathdoc},
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     number = {1},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2019_24_1_a3/}
}
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Vladyslav Bogun. Almost sure asymptotic expansions for profiles of simply generated random trees. Teoriâ slučajnyh processov, Tome 24 (2019) no. 1, pp. 49-63. http://geodesic.mathdoc.fr/item/THSP_2019_24_1_a3/