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November 1997 Some width function asymptotics for weighted trees
Mina Ossiander, Ed Waymire, Qing Zhang
Ann. Appl. Probab. 7(4): 972-995 (November 1997). DOI: 10.1214/aoap/1043862421

Abstract

Consider a rooted labelled tree graph $\tau_n$ having a total of n vertices. The width function counts the number of vertices as a function of the distance to the root $\phi$. In this paper we compute large n asymptotic behavior of the width functions for two classes of tree graphs (both random and deterministic) of the following types: (i) Galton-Watson random trees $\tau_n$ conditioned on total progeny and (ii) a class of deterministic self-similar trees which include an "expected" Galton-Watson tree in a sense to be made precise. The main results include: (i) an extension of Aldous's theorem on "search-depth" approximations by Brownian excursion to the case of weighted Galton-Watson trees; (ii) a probabilistic derivation which generalizes previous results by Troutman and Karlinger on the asymptotic behavior of the expected width function and provides the fluctuation law; and (iii) width function asymptotics for a class of deterministic self-similar trees of interest in the study of river network data.

Citation

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Mina Ossiander. Ed Waymire. Qing Zhang. "Some width function asymptotics for weighted trees." Ann. Appl. Probab. 7 (4) 972 - 995, November 1997. https://doi.org/10.1214/aoap/1043862421

Information

Published: November 1997
First available in Project Euclid: 29 January 2003

zbMATH: 0890.60082
MathSciNet: MR1484794
Digital Object Identifier: 10.1214/aoap/1043862421

Subjects:
Primary: 60J55 , 60J70 , 60J80 , 60J85

Keywords: branching process , Brownian excursion , Local time , occupation time , self-similar tree , width function

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.7 • No. 4 • November 1997
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