The Annals of Probability

Tightness for a family of recursion equations

Maury Bramson and Ofer Zeitouni

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Abstract

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 615-653.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099923

Digital Object Identifier
doi:10.1214/08-AOP414

Mathematical Reviews number (MathSciNet)
MR2510018

Zentralblatt MATH identifier
1169.60020

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks 39B12: Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX]

Keywords
Tightness recursion equations branching random walk cover time

Citation

Bramson, Maury; Zeitouni, Ofer. Tightness for a family of recursion equations. Ann. Probab. 37 (2009), no. 2, 615--653. doi:10.1214/08-AOP414. https://projecteuclid.org/euclid.aop/1241099923


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